Marcus Nunes directed me to a recent post where John Cochrane advocated price level targeting. I agree that price level targeting is superior to inflation targeting. He then responded to questions about nominal GDP level targeting:

Many comments here and on twitter ask about a nominal GDP target. I’m not a fan, for three reasons. 
First, just what does the Fed do to hit a nominal GDP target? That objection is common to the price level target, but it’s an important point. Nominal GDP targeting advocates seem to think it solves the whole conundrum of just how do interest rates affect inflation. No, it’s just a different centering point of the Taylor rule. Raise interest rates if nominal GDP growth is high, lower it if low. 
Second and more to the point, it assumes that “potential,” “supply” or “neutral” real GDP growth is constant or at least slow moving and known. If potential grows 2% real and you want 2% inflation, then the nominal GDP growth target is 4%, and the idea is that you let the economy take care of the split between real and nominal in the short run. I’m of the view that there is a lot more high frequency movement in “potential” than commonly thought, so even if the Fed achieved steady nominal GDP growth, there would be needless inflation volatility or needless deviation from neutral real growth. Real GDP grew 4% 1950-2000 and 2% since then. How long does it take the target to adapt to this sort of thing? 
The idea is that the Fed isn’t smart enough to separate nominal GDP growth to real growth and inflation, so let that be endogenous. Of all the problems of monetary policy, this doesn’t seem like the worst to me. 
Third, I like the clarity of a price level target. Even if you make it level, not growth, of nominal GDP, it’s muddy just how much inflation you should expect when borrowing money, financing a project, etc. Keep the units pure. Moreover, if you don’t like price index measurement issues, wait until you look in to GDP measurement issues. GDP is not consumer surplus. 
None of those objections are persuasive.
1.  The first point is correct, but (as Cochrane indicates) applies equally well to price level targeting (which he supports).  I favor employing NGDP futures contracts to guide policy.  I seem to recall that Cochrane once spoke positively of using CPI futures contracts in monetary policy.  
2. The second complaint has been made many times by NGDP critics, and refuted many times by NGDP proponents.   I’m not sure if Cochrane is familiar with that literature.  NGDP targeting does not make any assumptions about potential GDP growing at a constant rate.  Proponents see the variation in inflation due to RGDP fluctuations as a feature, not a bug in the NGDP regime.  (One possible exception is changes due to population growth—arguably it would be better to target NGDP per capita, or per adult.)
In addition, at the level of individual prices there might actually be more volatility with a price level target than with a NGDP target.  Consider an oil supply shock that raises oil prices by 50%.  Assume oil is 2% of the CPI.  Under a price level target you are forced to reduce non-oil prices by 1%.  That’s thousands of needless price changes—menu costs.  Under a NGDP target you would likely get a reduction in real GDP and a rise in the price level, leaving non-oil prices little changed.  Many non-oil prices (like haircuts) are tied to nominal wages, which tend to be pretty stable under a NGDP targeting regime.  When there’s an oil shock it’s easier to reduce real wages by allowing a slight rise in the price level, rather than squeeze nominal wages (and NGDP) lower at the cost of unemployment.
And this doesn’t even account for the issue of hedonics.  The CPI doesn’t measure the change in the average price of products, it measures changes in quality-adjusted prices.  If you have quality rising at 1%/year, you’d need the average nominal price of products to also rise at 1%/year to keep the measured CPI stable.  
So price level targeting is not the policy that does the best job of reducing “menu costs”; indeed NGDP does better when there are supply shocks.
3.  The third point is also incorrect—it’s easier to measure NGDP than inflation.  To measure (overall) inflation accurately you need to measure both nominal and real GDP.  Let’s say Tesla produces 2 million cars (final goods) at an average price of $60,000.  Then using the final goods approach to NGDP you could say that Tesla contributes $120 billion to NGDP.  But how does Tesla affect the price level?  To determine its impact on inflation you must measure the quality change over time in a Tesla car.  But what does “quality change” even mean?  The government says modern TVs are 99% cheaper than the TVs of 1960, mostly due to quality improvements.  What does that mean?  Does it mean I laugh 100 times as much watching Seinfeld on a modern TV as I would watching it on a 1960-era TV? I like modern TVs, and I agree they are vastly better—but 100 times better?  By what metric?  What does the term “better” even mean?  Let’s be honest, quality estimates are pulled out of thin air by government bureaucrats that have no objective method of measuring utility.  
I have been promoting NGDP targeting for decades.  I have yet to see any NGDP critic address the actual arguments being made by proponents of NGDP targeting.  I am certainly open to counterarguments—indeed I suspect NGDP targeting is not the optimal policy.  But it’s far better than what the Fed is currently doing.  We had a massive undershoot of NGDP in 2008-09 and a massive overshoot in 2021-23.  And we can all see what happened.
PS.  Under the value added approach to GDP, you’d add Tesla’s value added to the value added of its suppliers.
PPS.  If Cochrane insists on targeting a price index, I’d suggest he switch from price level targeting to nominal wage level targeting.  That’s an idea I could support.