Heat a medium sauce pot over medium-high heat. Once hot, add olive oil. Add diced onion and diced carrot to the hot olive oil. Saute until onion is softened, about 6-8 minutes. Add minced garlic and saute a minute more, until garlic is golden.
Add all ingredients except for the BBQ sauce: ketchup, mustard, brown sugar, chili powder, Worcestershire sauce, hot sauce, salt, vegetable broth. Stir to combine. Bring to a boil over high heat. Once boiling, lower to a simmer over medium-low. Cover and let simmer until lentils are cooked through, about 15-20 minutes.
Once lentils are cooked through, turn off the heat and stir in ½ cup barbecue sauce. Add additional sauce as desired.
Serve on burger buns with optional toppings. Refrigerate in a closed container for up to 5-6 days.
How to freeze: Add cooled (it should no longer be steaming) lentil sloppy joe filling to a freezer safe container (these are the kind that I use) and freeze for up to 3 months. Let thaw in the fridge before using or heat up in the microwave or stove for quick use.
Storage and reheating: Keep refrigerate in a closed container for 3-4 days. Reheat on the stove over medium heat or in the microwave in about 2 minutes.