Finely dice the onion, pepper, and carrots. Set aside.
Heat a 12-inch oven safe skillet over medium heat and add the ground beef.
Crumble the beef as it begins to brown. When about half of the beef is browned, add the onion, pepper, and carrots to the skillet. Continue cooking, stirring often, until the beef is cooked through.
Drain the grease from the skillet.
Whisk together the ketchup, mustard, brown sugar, vinegar, Worcestershire sauce, salt, and pepper. Pour this mixture into the skillet with the beef.
Cook over low heat, stirring occasionally, for 15 minutes.
Add the corn to the mixture and stir well.
Cut a slit in the film over the mashed potatoes and microwave for 2 minutes.
Add the mashed potatoes, grated cheddar, and chives to a mixing bowl and stir to combine.
Spread the mashed potatoes evenly over the sloppy joe mixture.
Place skillet in a preheated 350 degree oven and bake for 15 minutes.