Roast chiles (except for tomatillos) over a flame on the stove or on the grill until the skins are charred black on all sides.
Roast tomatillos in a dry skillet for 10 minutes, or until charred. Cool for 10 minutes, and then smash with a potato masher or fork until roughly chopped. Set aside.
Place chiles in a paper bag, seal well, and let rest of 20 minutes. Peel away black skin of chiles and discard seeds (unless you prefer SUPER hot) and stem; rinse well. Slice chiles into strips. Set aside.
Make a roux – melt butter over medium heat. Add flour and whisk well until combined. Stirring constantly, cook until roux is the color of peanut butter, about 7 minutes. Remove from heat and set aside.
Cut pork into 1 inch cubes. In a medium pot, heat ½ tablespoon canola oil over medium high heat. Working in two batches, fry pork until browned on all sides, adding ½ teaspoon more oil to the pot between batches.
Return all meat to the pot, cover with 6 cups of water and ½ teaspoon salt and bring to a simmer. Continue to simmer for 30 minutes.
Meanwhile, heat remaining oil in a large pot over medium high heat. Add onions; cook for five minutes or until just translucent. Add garlic and carrots; cook for another five minutes. Add ½ teaspoon of salt, cumin and pepper; allow to “toast” for 5 minutes more.
Add potatoes, pork and broth to the big pot; stir well to combine. Bring to a boil; turn down heat and simmer for 20 minutes.
Add remaining salt (1 teaspoon), broth, roux and ¼ cup cilantro to the pot; stir well to combine. Turn up heat and boil for five minutes, until the broth thickens.
To serve, ladle into bowls and top with remaining cilantro, queso, tomatoes, and radishes. Serve immediately with rice or tortillas.