1. Place beef chunks on baking sheet in single layer. Freeze meat until very firm and starting to harden around edges but still pliable, about 35 minutes. While meat is in freezer, heat 2 tablespoons vegetable oil in 12-inch skillet over medium-high heat until just smoking. Add mushrooms, 1/4 teaspoon salt, and 1/4 teaspoon pepper; cook, stirring frequently, until browned, 5 to 7 minutes. Add shallot and thyme and cook until fragrant. Remove skillet from heat and stir in dry sherry. Set aside.

2. Place one-quarter of meat in food processor and pulse until finely ground into ¹/16-inch pieces, about 35 one-second pulses, stopping and redistributing meat around bowl as necessary to ensure beef is evenly ground. Transfer meat to second baking sheet. Repeat grinding with remaining 3 batches of meat. Spread meat over sheet and inspect carefully, discarding any long strands of gristle or large chunks of hard meat or fat.

3. Adjust oven rack to middle position and heat oven to 300°F. Drizzle melted butter over ground meat and add 1 teaspoon pepper. Gently toss with fork to combine. Divide meat into 4 lightly packed balls. Gently flatten into patties 3/4-inch thick and about 4 1/2 inches in diameter. Refrigerate patties until ready to cook. (Patties can be refrigerated, covered, for up to 1 day.)

4. Season 1 side of patties liberally with salt and pepper. Using spatula, flip patties and season other side. Heat remaining teaspoon of oil in 12-inch skillet over high heat until just smoking. Using spatula, transfer burgers to skillet and cook without moving for 2 minutes. Using spatula, flip burgers and cook for 2 minutes longer. Transfer patties to rimmed baking sheet, top each with 1 ounce cheese, and bake until instant-read thermometer inserted into burger registers 125°F for medium-rare or 130°F for medium, 3 to 6 minutes.

5. Transfer burgers to plate and let rest for 5 minutes. Transfer to buns, top with mushrooms and Pub-Style Burger Sauce (if desired, see related recipe), and serve.