Heat up your pressure cooker over medium high heat. Press Saute button and Adjust button once to Saute More function. Ensure your pot is as hot as it can be. Instant Pot wait until indicator says Hot.
Season ground beef generously with kosher salt and freshly ground black pepper. Add a tablespoon of olive oil in pressure cooker and coat bottom of pot. Add ground beef in pressure cooker. The ground beef will begin to release moisture. At the 5 minute mark, drain and discard the ground beef juice.
Allow ground beef to brown. Stir occasionally until they are slightly crisped and browned, 5 to 7 minutes.
Add diced onion to browned beef and saute for 2 minutes. Add in diced bell pepper and minced garlic, then saute until fragrant (roughly a minute).
Add a tablespoon of brown sugar, a tablespoon light soy sauce, a tablespoon worcestershire sauce, and a half cup water and then fully deglace bottom of pot by scrubbing all the flavorful brown bits with a wooden spoon. Mix the brown bits with the liquid mixture.
Layer a tablespoon tomato paste, a teaspoon dijon mustard, and 3/4 cup ketchup on top. Do not mix. Close lid and pressure cook at high pressure for 5 minutes plus 10 minutes natural release. After 10 minutes, turn venting knob to venting position to release remaining pressure. Open the lid carefully.
On a skillet, toast burger buns for a few minutes over medium heat.
If the sloppy joe sauce is too runny, give it a quick stir and cook for another few minutes over medium heat (press saute button). Now is good time to add in the optional liquid smoke and cayenne pepper if desired. Taste and season with more kosher salt and black pepper if necessary.
Construct the sloppy joes with toasted hamburger buns. Serve with tater tots or your favorite side dishes.