Heat grill to medium-high. In a medium bowl, place turkey, jalapeno, cheese, breadcrumbs, mayonnaise, 1 teaspoon coarse salt, and 1/4 teaspoon pepper. Mix very gently with a fork to combine, being careful not to overwork meat (which would make burgers dense).
Gently form meat into 4 equal-size patties, 4 inches in diameter and 1 inch thick. Press thumb in the center of each patty to make an indentation approximately 1/2-inch deep (this helps keep burgers flat during grilling).
Moisten a folded paper towel with oil; grasp with tongs and rub over grates. Season burgers with salt and pepper; grill until cooked throughout, 5 to 8 minutes per side. Serve on buns (toasted on grill, if you like), with desired toppings.