Add split peas, stock, green pepper, onion, garlic, and mushrooms to a heavy bottomed soup pot over medium heat. Add water until liquid line is 2 inches (5 cm) above the mixture. Bring to a boil, reduce heat to a simmer and cover, cooking for about 30 minutes, or until the split peas are tender and have absorbed most of the liquid.
Meanwhile, whisk together tomato sauce, brown sugar, smoked paprika, garlic powder, onion powder, and chili flakes, if using. Scoop a bit of the mixture into a small cup and whisk in the corn starch. Set aside.
Drain excess liquid from the split pea & veggie mixture. Add tomato sauce mixture to the pot and stir to combine. Whisk in the corn starch slurry and add the Dijon mustard. Cook over low heat, stirring until slightly thickened. Add salt and pepper to taste.
Slice buns in half, place face open on a cooking sheet and broil in the oven for 2-5 minutes. Keep an eye on them so they don’t burn!
To serve, scoop split pea mixture onto buns, and enjoy!