In a small bowl, stir together; mayonnaise, mustard, ketchup, pickle brine, and 1/8 teaspoon cayenne pepper. Refrigerate until ready to use.
Using 2 tablespoons of oil, lightly oil two rimmed baking sheets; place mushroom caps gill side down. Brush mushroom caps with 2 tablespoons oil, season with salt and pepper. Place in preheated oven. Cook until tender, 30 to 35 minutes. Set aside until completely cool. Using a sharp knife, slice mushrooms in half horizontally, making sure to cut two even slices, without holes.
Meanwhile, make the mushroom filling. In a medium bowl combine: the cheeses, onion powder, garlic powder, 1/2 teaspoon flour, the egg yolks, and the cayenne pepper. Shape the filling into 8 equal patties, 1/2 inch smaller in diameter than the mushroom caps. Place prepared cheese patties on the gill side of each sliced mushroom cap, making sure to leave a 1/2 inch border. Top each with other side of mushroom pressing firmly around edges to seal filling inside. If it is difficult to seal mushroom patties filling remove some cheese mixture.
Place remaining flour in a wide dish. Place beaten eggs in another wide dish, place panko bread crumbs in a third wide dish, set aside.
Dredge the stuffed mushroom caps in the flour, dip in egg, coating them completely, then dredge in the panko. Repeat beading steps again with mushrooms. Refrigerate breaded mushrooms while forming burger patties.
Form burger patties:
Form hamburger patties into 8 (5-inch) wide patties, about 1/3 inch-thick, sprinkle with remaining teaspoon salt and remaining 1/4 teaspoon black pepper. Refrigerate.
In a medium sized heavy bottomed pot, fitted with a deep-frying thermometer, over medium-high, heat oil to 350 degrees F Line a baking sheet with paper towels. In batches, cook mushrooms, turning once, until deep golden brown on each side, about 2 to 3 minutes a side. Place mushrooms on prepared baking sheet to drain.
Heat a large cast iron skillet over medium-high, working in two batches place patties in skillet; using a large, sturdy spatula, firmly press burgers into the pan making sure they “smash slightly” and adhere to the pan. Cook, undisturbed, until bottom of burger is dark brown and crisp, 2 to 3 minutes. Flip patties, using the spatula to “smash” lightly again; top each burger with a cheese slice, cook until cheese has melted and burger patty is dark golden brown on other side, 1 to 2 minutes. Transfer burgers to a baking sheet. Repeat the process with the remaining burger patties.
Dividing evenly, top bottom buns with cheeseburgers, fried ‘shrooms, tomato, and lettuce. Spread tops of buns with ShackSauce. Close sandwiches. Serve Hot.