Preheat the oven to 170 degrees F. Grease a baking sheet with 2 tablespoons olive oil.
Pound the steaks to 1/2-inch thickness over plastic wrap. Sprinkle each with salt and pepper and place on the prepared baking sheet. Bake until it reaches an internal temperature of 80 degrees F, about 10 minutes.
In 2 large saute pans over medium-high heat, divide the remaining 2 tablespoons olive oil and the bacon fat between the pans and heat until shimmering. Add 2 steaks to each pan and cook for 3 to 4 minutes per side. Place the steaks onto a wire rack over a baking sheet. Top with fontina and then the prosciutto slices, evenly covering the cheese.
Adjust the oven to low broil and place the pan in the oven. Broil until the prosciutto is crisp, about 5 minutes. Remove and allow to rest 5 minutes, lightly covered with foil.
In each saute pan, evenly divide the shallots and cook for 2 to 3 minutes. Deglaze each with half the beef stock. Evenly divide the thyme and allow to reduce, 3 to 4 minutes. Combine the liquid into 1 pan, add the butter and adjust the seasoning, if desired.
Serve the steaks drizzled with the pan sauce and garnish with the parsley.