In a small saucepan, bring 2 cups of chicken broth to a boil. Remove from heat, and add the guajillo chiles, chiles de arbol, and the New Mexico Chiles. Cover the saucepan, and soak the chiles for 20 minutes.
Take the chiles out of the broth, and save the broth. Cut the top stems off, and remove the seeds by squeezing the peppers over a sink or bowl.
As the peppers soak, put the tomatoes in a dry skillet over medium-high heat, and roast until they are slightly charred on all sides. Remove as much of the tomato skin as possible and remove the cores. Put the tomatoes and the chiles into a food processor or blender along with ½ cup of the reserved chicken broth. Blend on high for about 1 minute, or until the peppers are finely chopped (strain through a mesh sieve, if desired). Set the marinade aside.
In a large bowl, salt and pepper the beef to taste. Lightly sprinkle the beef with flour, and stir with a wooden spoon until all of the beef is covered. In a large pot over medium-high heat, add 2 tablespoons of oil and the beef and sear on all sides. (You will need to work the beef in batches so the pot doesn’t get over crowded.) As the beef is finished browning, place it on a paper towel lined plate.
Once the beef is all seared, add the remaining tablespoon of canola oil to the pot. Pour in the white onion, and cook until it’s translucent, about 4 minutes. Add the garlic, and cook for one minute, or until fragrant. Pour in the oregano and cumin, and stir. Heat for about 30 seconds.
Return the beef to the pan along with the chili sauce, the reserved chicken broth, and the last 1½ cups of broth. Stir together and bring to a boil. Lower the heat to a simmer.
Use a ladle to spoon out about ½ cup of the stew liquid, and in a small bowl, mix it together with the cornstarch. Pour the mixture back into the pot and stir until combined. Cover the pot, and simmer the stew for 2 hours, stirring occasionally to avoid scorching. Stew is ready when the meat is tender and the sauce has thickened. Serve with rice and tortillas, and garnish with limes and cilantro.