Turkey Burgers with Maple Caramelized Onions

Heat the oil in a large skillet over medium-low heat. Add the onions and cook, stirring often, until they're softened and beginning to caramelize, 20 to 25 minutes. Add the maple syrup and cayenne and season with salt and black pepper to taste.
Cook the onions, stirring frequently, until shimmery and translucent, another 7 to 10 minutes. Transfer the onions to a plate and cover to keep warm. (You can cool, cover, and refrigerate the maple caramelized onions for up to several days. Gently warm them in a covered saucepan over low heat just before serving.)

Meanwhile, place a grill pan over medium-high heat or preheat a gas or charcoal grill.
Using your hands, in a large bowl mix the ground turkey, mustard, garlic, egg, parsley, salt, and pepper.
Gently form the mixture into four 6-ounce patties, being careful to press the turkey mixture together enough so that the turkey burgers remain intact but not so much they become compact and hard as hockey pucks. Brush each side of each burger with olive oil and season with salt and pepper. (You can cover and refrigerate the turkey burgers for up to 24 hours before cooking. And actually, refrigerating them isn't just a convenient make-ahead trick, it actually helps the turkey burgers hold together better during cooking. So we highly recommend sliding them in the fridge, even for as little as 30 minutes.)
Cook the turkey burgers, flipping them once, until cooked through, 6 to 7 minutes per side. (If using a grill pan, you'll probably need to turn the heat down to medium at some point.)
Place a slice of cheese on each burger, top with some of the maple caramelized onions, and use a spatula to slide the whole shebang to the waiting buns. (Any leftover maple caramelized onions can be covered and refrigerated for up to 1 month.)