Place the fennel in a storage container with a lid. In a small saucepan, bring about 1/2 cup vinegar, 1/4 cup water, 3 tablespoons sugar and 1 teaspoon salt to a boil and cook until the salt and sugar dissolve. Pour over the fennel, cover and refrigerate for at least 1 hour.
Preheat an outdoor grill or a grill pan to medium-high. In a skillet, heat 2 tablespoons EVOO , 2 turns of the pan, over medium-high heat. Add the bell peppers, red onion and sliced garlic; season with salt and pepper. Cook until tender, 7 to 8 minutes. Sprinkle the bell peppers with the remaining 1 teaspoon sugar and 1 tablespoon vinegar. Stir for a minute more and turn off the heat.
Season the ground pork and beef with salt, pepper, the paprika, fennel seeds, granulated garlic, granulated onion and sage; mix well. Form the meat into 4 oval burgers and drizzle with about 1 tablespoon EVOO. Add to the grill or grill pan and cook through, turning once, 10 to 12 minutes.
In a small saucepan, heat the butter and crushed garlic in a drizzle of EVOO over low heat to infuse the flavors, 2 to 3 minutes.
Preheat the broiler and lightly toast the rolls. Set the roll bottoms aside. Brush the roll tops with the garlic butter and sprinkle with the cheese. Broil until the cheese is lightly browned, about 1 minute.
Place the basil on the roll bottoms. Top with the peperonata, sausage patties, drained pickled fennel and roll tops.