Burger Spaghetti

Name: Burger Spaghetti
Ingredients: Kosher salt; 3 stale hamburger buns; 2 tablespoons extra-virgin olive oil; 1/2 cup whole milk; 1 pound ground beef; 1/2 cup finely chopped onion; 1/4 cup finely chopped dill pickle slices, plus pickle slices for serving; 1/4 cup ketchup; 1 tablespoon yellow mustard; 1 tablespoon Worcestershire sauce; Freshly ground pepper; 3 ounces sharp white cheddar cheese (half cut into 24 small cubes, half grated); 12 ounces spaghetti; 4 cloves garlic, sliced; 2 tablespoons tomato paste; Thinly sliced romaine lettuce, for topping; Diced tomato, for topping