In a large skillet over medium heat, melt 1 tablespoon butter. Add diced onion, green bell pepper, and garlic; sauté until softened, about 5 minutes.
Add ground beef to the skillet and cook until browned, breaking it up with a spoon, about 7 minutes.
Stir in ketchup, tomato paste, Worcestershire sauce, yellow mustard, and brown sugar. Reduce heat to low and simmer the mixture for 10 minutes, stirring occasionally, until thickened and saucy.
Meanwhile, spread remaining butter on one side of each Texas toast slice. Toast bread slices in a separate pan over medium heat until golden and crispy on both sides.
Spoon the hot sloppy joe mixture generously over the toasted Texas toast slices, piling it thickly.
Serve immediately on white round plates.