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In a medium bowl, mix bulgur with 1/4 teaspoon coarse salt and 1 cup boiling water. Cover bowl, and let sit until bulgur is tender (but still slightly chewy), about 30 minutes. Drain in a fine-mesh sieve, pressing to remove liquid.
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Place beans in a medium bowl; mash with a potato masher until a coarse paste forms. Add breadcrumbs, scallions, egg, carrot, cayenne, tahini, and bulgur. Season with salt and pepper, and mix to combine. Form mixture into 4 patties, each about 1 inch thick.
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In a large skillet, heat oil over medium-low. Cook patties until browned and firm, 5 to 8 minutes per side. Serve on English muffins with tahini mayonnaise and lettuce.
