In a large skillet over medium heat, melt 1 tablespoon of butter. Add diced onion, green bell pepper, and minced garlic. Sauté until softened, about 4 minutes.
Add ground beef to the skillet. Cook, breaking up the meat with a spoon, until browned and cooked through, about 6 minutes. Drain excess fat if needed.
Stir in ketchup, tomato paste, Worcestershire sauce, brown sugar, yellow mustard, salt, and black pepper. Reduce heat to low and simmer gently, stirring occasionally, until thickened and saucy, about 8 minutes.
Meanwhile, butter one side of each Texas toast slice with remaining butter. Toast in a separate pan or under the broiler until golden and crispy, about 2-3 minutes per side.
To serve, pile the hot sloppy joe mixture thickly on the toasted side of each slice of Texas toast. Serve open-faced on simple white plates.