Heat grill to medium (350° to 450°). Mix bison, oats, 2 tbsp. ale, the salt, and pepper in a large bowl until just combined. Form into four 1/2-in. patties with a slight depression in the center of each to help them cook evenly. Brush with oil.
Whirl garlic in a food processor until minced. Add cheeses and remaining 2 tbsp. ale and whirl until smooth.
Grill burgers, turning once, until cooked the way you like, 6 minutes total for medium. In last few minutes, grill split buns, turning once, about 2 minutes total.
Arrange burgers on bottom of buns and top with sprouts. Spread bun tops with cheese mixture.