Bring water and brown rice to a boil in a saucepan. Reduce heat to medium-low, cover, and simmer until rice is tender and liquid is absorbed, 45 to 50 minutes. Remove from heat to cool slightly.
Stir barbeque sauce and molasses together in a bowl.
Mash beans in a bowl; stir in 3 tablespoons of the barbeque sauce-molasses mixture. Add rice, onion, beets, beet juice, black pepper, and salt; stir to combine. Refrigerate until firm, 15 to 30 minutes.
Form burger mixture into 4 equal-size patties.
Heat oil in a skillet over medium heat. Add burgers and cook for 2 minutes. Turn over and brush with remaining barbeque sauce-molasses mixture. Cook until golden and heated through, about 2 minutes more.