Heat grill to high. Cut nectarines into thick slices. Coat slices with olive oil and place on hot grill. Do not cover. Grill until bottom side is lightly charred; flip and lightly char other side. Remove slices from grill. Do not allow to caramelize or cook through.
Slice brioche into eight 1/2-inch-thick slices. Press slices between 2 sheet pans to flatten. Top 4 bread slices each with slice of fontina cheese. Cover each with slice of plain bread; repeat pressing process to make 4 thin sandwiches. Heat skillet over medium-low; add 1 tablespoon butter. Toast both sides of sandwiches until browned and crisp and cheese is melted, adding 1 tablespoon butter each time you flip, coating each side with butter. Remove sandwiches from skillet; keep warm.
Slice shallot as thinly as possible. Add to hot skillet and cook just until tender; reserve. Top 2 thin sandwiches each with a dollop of Wisconsin mascarpone cheese, half the grilled nectarines and 2 slices prosciutto. Toss arugula with a bit of lemon juice to coat. Place over prosciutto and top with reserved shallots. Place remaining 2 fontina-filled thin sandwiches over to close and press. Serve immediately.