Saute shallot in olive oil over medium-low heat until softened. Set aside to cool.
To make patties combine ground sirloin, ground chuck, salt, pepper, Worcestershire sauce, mustard, sage, and sauteed shallot in a large bowl. Mix well until all ingredients are thoroughly combined. Divide mixture into 6 equal portions and form patties (6 ounces each). Grill patties (in grill pan) over medium-high heat for 4 minutes on each side for medium rare.
While patties are cooking, prepare buns. Whisk eggs in a bowl with milk, salt, and pepper. Dip Portuguese muffin halves in mixture 1 at a time (place each on a plate after dipping). Heat 1 tablespoon butter in saute pan over medium heat. Place 3 muffin halves into pan and fry, flipping once until both sides are golden brown. Place cooked muffins onto plate and cover with foil. Repeat process using 1 tablespoon butter for every batch of 3 muffin halves until all have been cooked.
When patties have been cooked, remove the patties from the grill pan and rest on a half-sheet pan. Top with cheese slices and tent with aluminum foil. Repeat process with the remaining patties.
To assemble the burgers, place 1 patty (with cheese) on bottom half of the Portuguese muffin. Top with prosciutto, spread with 1 tablespoon of sauce, and add top of muffin. Serve.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.