Found it! Opened up the carb to clean and found a ton of crap in the bowl and packed around the screen filter. Apparently, the gunk loosened and fell back into the bowl when not running. Then, once the mower was started, the gunk began collecting arounnd the filter again until it gathered so much that it eventually choked out the fuel supply.
After a thorough carb cleaning (and resetting the float from just over 9/16" to 7/16") it's running like a champ!
After a thorough carb cleaning (and resetting the float from just over 9/16" to 7/16") it's running like a champ!