Preheat the grill to high heat. Portion and form the ground beef into six hamburger patties, making an indentation in the centers. Salt and pepper both sides of the patties.
Slice the watermelon into 1/3-inch slices. Use a biscuit cutter to cut watermelon circles.
Grill the hamburgers for 3-5 minutes per side. Add the fontina slices to the top of each patty 1 minutes before removing from the grill. Tent with foil to keep warm.
Toast the buns on the grill for 30-60 seconds, until char marks form.
Layer the bun with 2-3 large basil leaves, a watermelon circle, the patty with cheese, and top with a large spoonful of salsa verde. Serve immediately.