In a medium bowl, mix ground turkey, jalapeño and shallot until combined well. Divide mixture into 4 equal portions and shape into patties. Season patties with salt and pepper on both sides.
Grill patties over medium heat, turning once, until cooked all the way through, about 4 minutes per side. Put buns, sliced side down, on grill to toast lightly.
Assemble burgers on buns with cheese and avocado slices. Add fresh tomato, lettuce and onion and mustard, mayonnaise and ketchup, if desired.