Peter Schickele?

 The credit to this discovery goes to the "Why Are You Laughing" podcast, which you can listen to here.

In 1990, the nominees for the Grammy for Best Comedy Album were:

• Sandra Bernhard - Without You I'm Nothing
• Erma Bombeck - Motherhood: The Second Oldest Profession
• Andrew Dice Clay - Dice
• Sam Kinison - "Wild Thing"
• Peter Schickele - P.D.Q. Bach: 1712 Overture and Other Musical Assaults

Winner: Peter Schickele

In 1991, the nominees for the Best Comedy Album Grammy were:

• Bob Elliott & Ray Goulding - The Best of Bob and Ray: Selections from a Career, Vol. 4
• Garrison Keillor - More News from Lake Wobegon
• Various artists - The Best of Comic Relief '90
• Jonathan Winters - Jonathan Winters into the '90s
• Peter Schickele - P.D.Q. Bach: Oedipus Tex and Other Choral Calamities

Winner: Peter Schickele

1992, Best Comedy Album:

• Erma Bombeck - When You Look Like Your Passport Photo, It's Time to Go Home
• George Carlin - Parental Advisory: Explicit Lyrics
• Garrison Keillor - Local Man Moves to the City
• Jackie Mason - Brand New
• Peter Schickele - P.D.Q. Bach: WTWP Classical Talkity-Talk Radio

Winner: Peter Schickele

1993:

• George Burns - An Evening with George Burns
• Rita Rudner - Naked Beneath My Clothes
• Jonathan Winters - Jonathan Winters is Terminator 3
• "Weird Al" Yankovic - Off the Deep End
• Peter Schickele - P.D.Q. Bach: Music for an Awful Lot of Winds and Percussion

Winner: Peter Schickele

Peter Schickele won 4 times in 4 straight years, against a murderer's row of talent: George Burns, Jonathan Winters, Weird Al, Jackie Mason, George Carlin, Sam Kinison, Various Artists... this guy beat them all.

Have you ever heard of him? Have you ever heard of someone who's heard of him? Isn't this guy on the Mt. Rushmore of comedy? How did this happen? Is this guy the most critically acclaimed yet underrated comedian in history? Well, let's take a look at his act.

Okay, well that's not great. But it's also pretty visual and he got his awards from his "P.D.Q. Bach" albums. We need to examine those.

Wikipedia:

Schickele developed an elaborate parody around his studies of P.D.Q. Bach, the fictional "youngest and the oddest of the twenty-odd children" of Johann Sebastian Bach.

Okay, well I can't find his albums on youtube directly but here's one of the songs on an album:

Comedy?! Man, awards mean nothing. People still venerate them and I used to do that myself but they are literally meaningless.