Add the beans to a large mixing bowl. Gently pat beans dry with a paper towel. Using the back side of a fork or potato masher, mash beans until smooth and pasty.
Heat a small skillet over medium heat.
When hot, add the oil, onion and garlic. Sauté 3 minutes then transfer to the bowl with the beans.
In a small bowl, add the paprika, cumin, chili powder, salt and pepper. Mix until combined then add to the large bowl.
Using the same small bowl, mix the BBQ sauce and molasses until thoroughly combined.
Add 3 tablespoons of the mixture to the large bowl and set aside the rest for basting during cooking
In a food processor fitted with a metal blade, process oats until fine.
Add the oats to the large bowl with the beans, veggies and sauce.
Add the rice, beets, beet juice and egg and mix everything until combined.
Using your hands, form the mixture into 4 patties, about 5 ½ ounces each. Place patties on a freezer safe plate and freeze for 30 minutes. Alternatively, you can put them in the refrigerator for 2 hours or more.
Heat a large flat skillet over medium heat. Spray with oil and, using a spatula, transfer patties to the skillet.
Cook the patties for 5 minutes, carefully flip with the spatula, baste with the remaining sauce and cook an additional 5 minutes. Add cheese during the last 2 minutes, if using.
With the spatula, place each patty on a bun and top with your desired toppings.