Cut a slit in a thick side of each mushroom cap to form a pocket. (Do not cut them in half.) Carefully stuff 1/4 cup of mozzarella into each mushroom pocket with your fingers or a spoon, pressing and flattening the cheese as you stuff. (Do not overstuff; see Note.)
Stir together the flour and ranch dressing mix in a small bowl. Put the eggs in a second bowl and the panko in a third bowl. Dip both sides of each mushroom in the flour mixture; then dip in the eggs, and then dredge in the panko. (Make sure the mushrooms are thoroughly coated in the panko.)
Preheat the oven to 375° F. Heat 1/2 inch of the oil in a medium saucepan over medium until splattering. (If a few panko crumbs dropped in the oil turn golden quickly, you’re ready to fry.) Fry the mushrooms, flipping them frequently and keeping a close eye on them, until they turn golden and the cheese starts to melt, 4 to 5 minutes. Transfer to an aluminum foil-lined baking sheet, and bake until the cheese is fully melted, about 5 minutes. Transfer to a paper towel-lined plate, and season with 1/2 teaspoon of the salt 1/4 teaspoon of the pepper.
Slather the top and bottom halves of each bun with 1 tablespoon Thousand Island Dressing. Layer each bottom half with the pickles, lettuce, and 2 tomato slices. Sprinkle with the remaining 1/4 teaspoon salt and 1/8 teaspoon pepper. Top with the fried mushrooms and the bun top.
These burgers can be a little messy. After stuffing the burgers with the mozzarella, chill them for 30 minutes to let them set up.