In a large skillet over medium heat, sear the ham steak until browned on both sides. Transfer it to a cutting board, dice it into small cubes, and leave the flavorful drippings in the skillet for extra flavor.
Add the ground beef, Italian sausage, and diced onion to the same skillet. Cook over medium-high heat for 8 to 10 minutes, breaking the meat apart as it cooks, until the beef and sausage are browned and fully cooked through. Drain off excess grease if needed.
Stir in the minced garlic, tomato paste, paprika, cumin, chili powder, salt, and black pepper. Cook for 1 to 2 minutes, stirring well, until the tomato paste darkens slightly and the seasonings coat the meat evenly.
Add the salsa and seared diced ham steak to the skillet. Let the mixture simmer for 2 to 3 minutes so the flavors come together and the filling stays juicy without being watery.
Warm the flour tortillas in the microwave, in a dry skillet, or over a low flame until they are soft and flexible. Lay each tortilla flat, then add cooked rice down the center. Spoon the meat mixture over the rice and top with shredded cheese.
Fold in the sides of each tortilla, then roll tightly into burritos. Place the burritos seam-side down in a hot skillet over medium heat and toast for 2 to 3 minutes per side, until golden brown and lightly crisp. Serve warm with sour cream, salsa, guacamole, cilantro, or hot sauce.