Melt butter in a skillet over medium-high heat. Saute onions until translucent, about 5 minutes. Add sugar, stir, and saute until onions are softened and lightly browned, about 5 minutes more. Transfer onions to a food processor, leaving as much butter in the pan as you can.

Fry livers in the hot butter until lightly browned on 1 side, about 4 minutes. Turn and continue frying until slightly pink in the center, 3 to 4 minutes more. Add livers to the food processor with the onions.

Pour bourbon into the same skillet and bring to a boil while scraping the browned bits off the bottom of the pan with a wooden spoon. Boil until thickened and reduced by half, about 6 minutes. Pour liquid into the food processor.

Combine eggs, thyme, salt, and pepper in the food processor with the liver mixture. Process until smooth.