Prepare a medium-hot fire in a charcoal grill with a cover, or preheat a gas grill to medium-high.
To make White Gazpacho Mayonnaise, in the bowl of a food processor combine almonds, grapes, garlic, sherry vinegar, and olive oil; puree. Add mayonnaise and pulse until well combined. Refrigerate until serving.
To make the Madeira mushrooms, heat olive oil in a large fireproof nonstick skillet on the grill over medium-high heat. Add mushrooms to the skillet and cook for about 8 minutes, stirring occasionally. Add Madeira wine and continue to cook until most of the wine has evaporated. Season, to taste, with salt and pepper. Set aside.
To make the patties, combine the beef, linguica sausage, salt and pepper in a large bowl. Handling the meat as little as possible to avoid compacting it, mix well. Divide the mixture into 6 equal portions and form into patties to fit the rolls.
Brush the grill rack with vegetable oil. Place the patties on the rack, cover, and cook, turning once, just until done, about 4 minutes on each side for medium-rare. During the last few minutes of cooking, top each burger with equal amounts of cheese and place the rolls, cut side down, on the outer edges of the rack to toast lightly.
To assemble the burgers, spread a generous amount of White Gazpacho Mayonnaise on the cut side of the rolls. On each roll bottom, place a slice of tomato, a cheese-topped patty, and Madeira mushrooms. Add roll tops and 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.