Bring the bulgur and 1 1/2 cups water to a boil in a small saucepan. Reduce the heat to medium-low, cover and simmer until tender, 10 to 12 minutes. Strain, and let cool completely.
Meanwhile, heat the oil in a small skillet over medium-high heat. Add the tomato paste, chili powder, cumin and cayenne, and whisk until the oil is yellow and the spices are fragrant, about 45 seconds. Let the chili oil cool to room temperature.
Whisk together the sour cream, lime zest and juice, a pinch of salt and 1 tablespoon water in a small bowl. Refrigerate until ready to serve.
Position an oven rack at the top of the oven, and preheat the broiler. Line a baking sheet with foil. Mash the beans, cooked bulgur, cheese, breadcrumbs, scallions, chili oil and 1/2 teaspoon salt with a potato masher in a large bowl until the mixture is well combined and holds together when squeezed. Form into 4 uniform patties about 3/4 inch thick.
Arrange the patties on the prepared baking sheet, and generously coat each with cooking spray. Broil until a golden-brown crust forms, 4 to 5 minutes. Remove the baking sheet from the broiler, flip each patty with a metal spatula and coat the tops with cooking spray. Broil until the tops are golden brown, 4 to 5 minutes more.
Halve, pit and slice the avocado. Put each patty on a muffin bottom and sprinkle with additional cheese if using; top with some sour cream-lime sauce, avocados and sprouts, and sandwich with the muffin top.
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