In a large skillet over medium heat, melt 1 tablespoon butter. Add diced yellow onion, green bell pepper, and minced garlic. Sauté until softened, about 5 minutes.
Add ground beef to the skillet. Cook, breaking it up with a spoon, until browned and no longer pink, about 7-8 minutes.
Drain excess fat if necessary. Stir in ketchup, tomato paste, Worcestershire sauce, yellow mustard, brown sugar, salt, and black pepper.
Reduce heat to low and simmer the mixture, stirring occasionally, until thickened and saucy, about 15 minutes.
In a separate pan, melt remaining 1 tablespoon butter over medium heat. Toast the Texas toast slices until golden brown on both sides.
To serve, spoon the sloppy joe mixture generously over two slices of toasted Texas toast on a simple white plate.