1. Preheat oven to 500 degrees. In a large pot of boiling salted water, cook pasta according to package instructions. Reserve 2 1/2 cups pasta water; drain.

2. Meanwhile, in a large bowl, combine breadcrumbs, milk, 3/4 teaspoon salt, and 1/4 teaspoon pepper. In a large heavy pot, melt butter over medium. Add shallots and cook, stirring occasionally, until softened, about 5 minutes. Remove from heat; transfer 1/4 cup shallots to breadcrumb mixture, along with beef and stir until just combined (do not overmix). Form into 1-inch balls and place, 1 inch apart, on a rimmed baking sheet. Bake until browned and cooked through, about 7 minutes, turning halfway through.

3. Add mushrooms to pot with remaining shallots and cook, stirring, over medium until mushrooms are softened, 5 minutes. Season with salt and pepper. Add reserved pasta water, scraping up browned bits with a wooden spoon, and bring to a rapid simmer. Cook until liquid is reduced by half, about 6 minutes. Remove from heat and stir in sour cream and 2 tablespoons dill. Add meatballs and toss to coat. Divide pasta among 4 bowls and top with meatballs, sauce, and remaining 2 tablespoons dill.