This is classic chili – more or less Tex-Mex style. If you make it with the whole ancho chile pods (sometimes called pasilla chile pods on the West Coast), use coarse-ground meat (the classic “chili grind” in the Southwest), brown it well, and add fresh-ground spices; it’ll be one of the best chilis you’ve tasted.
The difference between chili and chile? Chili is the dish; chile is the pod. Why not use regular chili powder? Because it usually contains dried spices (including salt and sugar), which you may not like. If chili powder’s all you’ve got, use 4 tablespoons, leave out the cumin, and go easy on the salt. Chile pods and pure ground chile are available in all Mexican grocery stores.
Heat large deep skillet or a large heavy pot (like a 6- to 9-quart Dutch oven) over medium. When hot, toast chiles one by one: open flat and press down with spatula until chile releases aroma and toasts lightly - 10 to 15 seconds. Flip and toast other side same amount of time. Transfer to a small bowl. When all are done, cover with hot tap water and lay a plate on top to keep the chiles submerged.
Raise heat under pan to medium-high. Add bacon drippings or oil, then add the meat and onion. Break up meat with spoon or spatula as it cooks and browns—total cooking time about 10 minutes. Remove from the heat. If there is lots of rendered fat, tip it off and discard.
Drain chiles, discarding water. Place in food processor fitted with steel blade. Add garlic, tomatoes with their juices and cumin. Secure lid and process until smooth. Set medium-mesh strainer over meat pot. Pour in chile mixture and press through.
Return pan to medium-high heat. Stir 5 minutes to cook chile mixture. Stir in 2 cups water and 1 1/2 teaspoons salt. Reduce heat to medium-low. Simmer 45 minutes.
Sprinkle corn meal or masa harina over chili and stir. Stir in drained beans if using. Simmer 5 minutes. Taste and season with more salt if you think necessary. Ladle chili into bowls. Scoop cheese and green onion into small serving bowls. Pass separately for guests to add as much as they want.