Heat large skillet over med-low heat. Add garlic, cook until tender and skin begins to blacken, turning occasionally, about 18 minutes. Remove from skillet. Add chilies to skillet, flatten with spatula. Toast over med-low until beginning to darken, 2 min per side. Place in bowl. Add enough hot water to bowl to cover chilies. Soak until very soft, about 30 minutes.
Drain chilies. Peel garlic. Pour 2 cups broth in blender. Add chilies and garlic, puree. Heat oil in heavy med saucepan over med-high heat. Add chili mixture, simmer until thick, stirring often, about 5 minutes. Add 1 1/2 cups broth, simmer over med-low heat until sauce is reduced to 1 1/2 cups, stirring often about 25 minutes. Add sugar.
Preheat oven to 350 degrees. Spread 1/4 cup sauce over bottom of 9x13 inch baking dish. Cover sauce with 4 tortillas. Top with beans, then 1 cup cheese. Cover cheese with 4 tortillas. Pour half of remaining sauce over tortillas. Sprinkle 2 cups cheese over sauce. Drizzle with half of sour cream mixture. Layer 4 tortillas over sour cream and cheese. Pour remaining sauce over tortillas. Sprinkle with remaining cheese and drizzle with remaining sour cream. Cover with foil. Bake 15 minutes. Uncover; bake until casserole is bubbling around edges and cheese melts, about 20 minutes longer.