Heat a gas or charcoal grill for medium-high heat cooking.
Toss the onions with the olive oil and season with salt. In a skillet over medium-high heat, caramelize the onions until softened and golden brown, about 20 minutes. Remove the a plate or bowl to cool completely.
Use a filet knife to make a small slit in the side of each filet. Then use the tip of the knife to make a small pocket in the filet just below the slit. Stuff each filet with caramelized onions, cubed fontina, basil and a piece of prosciutto. Add the garlic and more onion. Season the steaks all over with kosher salt.
Grill the steaks directly over medium high heat for 2 to 3 minutes per side, flipping regularly, until the cheese is melted (it may ooze out) and the steaks reach 125 degrees F internally, about 15 minutes total.
Rest the steaks for about 10 minutes, slice for serving and top with more basil, flaky salt, and olive oil just before serving.