Let 4 tablespoons unsalted butter sit at room temperature until softened.
Heat 1 tablespoon olive oil in a large skillet over medium heat until shimmering. Add 1 pound lean ground beef and cook until cooked through and browned. Season with 1 teaspoon kosher salt, 1 teaspoon garlic powder, and 1 teaspoon onion powder. Cook until fragrant, about 1 minute more.
Add 1/2 cup water, 1 tablespoon packed brown sugar, and 1 teaspoon Worcestershire sauce. Stir to combine and cook until saucy. Remove the pan from the heat.
Butter one side of 8 slices sourdough bread (1/2 tablespoon each). Flip 4 of the slices over. Sprinkle each with 1/2 cup shredded sharp cheddar cheese. Top each with 1/2 cup sloppy Joe mixture, then sprinkle each with another 1/2 cup of the cheese.
Close the sandwiches with remaining bread slices buttered-side up. Heat a large skillet over medium-high heat. Cook 2 sandwiches at a time: Add to the pan and cook until golden-brown and cheese is melted, about 4 minutes per side. Transfer to a cutting board and cut in half.