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 4 1/2 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. Spread 1 tablespoon Fig Spread on top of each bun. Transfer burgers to buns. Top burgers with crispy shallots, and serve.