- 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 beef is in freezer, heat 1/2 cup vegetable oil and shallots in medium saucepan over high heat; cook, stirring frequently, until shallots are golden, about 8 minutes. Using slotted spoon, transfer shallots to towel-lined plate, season with salt, and set aside.
- 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.
- Adjust oven rack to middle position and heat oven to 300 degrees. 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 4½ inches in diameter. Refrigerate patties until ready to cook. (Patties can be refrigerated, covered, for up to 1 day.)
- Season 1 side of patties liberally with salt and pepper. Using spatula, flip patties and season other side. Heat 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 crumbled blue cheese, and bake until instant-read thermometer inserted into burger registers 125 degrees for medium-rare or 130 degrees for medium, 3 to 6 minutes.
- Transfer burgers to plate and let rest for 5 minutes. Transfer to buns, Top with Pub-Style Burger Sauce (see related recipe) and crispy shallots, and serve.
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