In a medium skillet, heat oil over medium heat until shimmering. Cook the shallots, stirring for 8 minutes, or until they begin to soften and colour slightly. Reduce the heat to low, cover, and cook, stirring occasionally, for 10 minutes, or until the shallots become dark brown and very soft. (If the pan becomes too dry during cooking, add 1 to 2 tablespoons water.) Stir in the vinegar.
Meanwhile, in a large bowl, combine the turkey, blueberries, bread crumbs, lemon juice, mustard, garlic, salt, and pepper. Form into two ¾" (1.2cm) thick patties. Preheat a grill pan over medium-high heat until smoking. Coat with cooking spray. Grill the burgers, turning once, for 8 minutes, or until grill marks from on both sides, a thermometer inserted in the centre registers 75°C/165°F, and the meat is no longer pink.
Toast the buns in a toaster oven or under the broiler.
Add the cheese to the burgers, cover and cook for 1 minute, or until melted. Top the bottom half of each bun with a lettuce leaf, tomato slice, burger, half of the caramelised shallots, and top bun.