Prepare the sauce first. Heat olive oil in a pan. Add shallots and cook for 1-2 minutes. Add tomato paste, broth, and red wine. Simmer for 2-3 minutes. Add brown sugar, salt, black pepper, and mustard. Simmer over low heat for 3-5 more minutes until the sauce has thickened and sugar has dissolved. Remove from heat, cover and keep warm or, store in the refrigerator and rewarm right before serving.
Heat olive oil in a medium saucepan. Chop pancetta and cook pieces. Once cooked through, scoop onto a paper towel-lined plate and set aside. Leave at least 1 tablespoon of rendered fat in the skillet.
Clean mushrooms and chop into small pieces. Add shallot to the pan and cook for 1-2 minutes. Add garlic and cook briefly. Next up are the mushrooms, Worcestershire sauce, and remaining seasonings. Cook for 4-5 minutes total. Add pancetta back to the skillet, stir through and remove from stove. Set aside at let cool for 15 minutes.
Remove any excess fat from rib-eye steaks. Cut into strips. Set the grinder to a medium grind and feed the meat through. Form ground steak into 8 equal-sized patties. Place filling into four of the patties. Do not overfill or they will break and fall apart on the grill. It's okay to have some leftover filling. Top with remaining four patties. Secure edges.
Preheat grill for medium-high heat.
Place patties onto grill and cook for 4-5 minutes per side or until the internal temperature reaches 165 degrees F. Toward the last minute or two of cook time, add cheese slices to burger patties. Brush hamburger buns with oil and place onto the grill until they have toasted. About 1-2 minutes. If you have a side burner on your unit, then use a skillet to fry onion rings. Otherwise, bake them in your oven and keep warm until it's time to serve.
Remove buns and patties from grill and assemble burgers. Serve immediately.