Step 1In a large bowl, combine the ground rib-eye and sausage with the crumbled bacon and diced tomatoes. Add the maple syrup, beef broth, beer, ketchup and egg, along with the reserved bacon fat. Mix gently, so as not to overwork the meat. Add the pepper, salt, paprika, oregano, crushed red pepper, chili powder, garlic powder, celery salt, cayenne pepper and bread crumbs, mixing gently until all of the ingredients are thoroughly combined. Cover the mixture and rest it in the refrigerator.
Step 2In a large pan of gently simmering water, poach the quail eggs, careful not to break the yolks or overcook. Drain and set aside to cool.
Step 3Remove the meat mixture from the refrigerator and divide it into 12 even portions. Form each portion into a small round patty, adding an indentation to the center of 6 patties for the poached quail eggs to nest.
Step 4Place a poached egg in each of the 6 indented patties, then place another patty over it to form the top, gently crimping the edges together so as not to crush the egg and break the yolk. Repeat to form 6 stuffed patties.
Step 5Prepare the grill. Place the burgers, top-side down, on the grill. Cook 7 to 10 minutes. then gently flip. Cook the other side 5 to 8 minutes. Remove from heat, cover and set aside to rest.
Step 6While the burgers are resting, toast the muffin halves. When toasted, remove from heat and set aside to cool a minute.
Step 7To assemble the burgers, place a handful of arugula on the toasted bottoms of the 6 muffins. Place a burger on top of each mound, bottom-side down. Top each burger with the other toasted muffin half. If preferred, slice in half before serving, or simply eat whole. Serve immediately.