Bring 3 cups beef broth to a boil. Put chiles in a blender and pour boiling broth over them. Let stand, mixing occasionally, until chiles are limp, about 10 minutes. Add onion, garlic, cumin, and oregano and whirl until very smooth.
Meanwhile, preheat oven to 325°. Sprinkle beef all over with salt and pepper. Pour oil into a large, heavy pot (not extremely wide) over medium-high heat. When hot, add roast and cook, turning as needed, to brown all over, about 10 minutes total. Pour chile mixture over meat and add enough beef broth to come a third to half of the way up roast. Bring to a boil, cover pot, and transfer to oven.
Bake until beef is very tender when pierced, about 3 1/2 hours, turning once. Transfer roast to a board. When cool enough to handle, remove string and pull meat into bite-size chunks, discarding excess fat.
Meanwhile, skim fat from chile sauce in pot. Boil sauce over high heat, stirring often, until reduced to about 4 cups. Add salt and pepper to taste. Return beef to sauce and simmer until heated through.
Note: Nutritional analysis is per serving.