Place chilies in a large tall-sided skillet over medium-high heat and gently heat until fragrant, about 2 to 3 minutes per side. Don't let them burn, or they'll turn bitter. Place chilies in a bowl, and cover them with very hot water until soft, about 20 to 30 minutes, turning once or twice.
Drain the chilies. Split the chilies open and remove stems and seeds. Place chilies in a blender with cumin, pepper, salt and 1/4 cup of the water. Puree the mixture, adding more water as needed, and occasionally scraping the sides of the blender jar, until smooth and forms a paste. Set the paste aside.
Return the skillet to medium-high heat, and add 2 tablespoons oil. When it begins to smoke lightly, add half of the beef and lightly brown both sides, about 3 minutes per side. Repeat with remaining beef, adding 2 more Tbsp oil to pan. Transfer browned beef to a bowl. Reserve the pan gravy.
Let the skillet cool to medium-low heat and add remaining 1 Tbsp oil. Add onion and garlic to skillet and gently sauté in hot oil 3 to 4 minutes, stirring occasionally. Add stock and 2 cups water to skillet, then whisk in the masa harina to avoid lumps. Stir in the reserved chili paste, scraping the bottom of the skillet with a spatula to loosen browned bits. Add the beef and reserved pan gravy, and bring mixture to a simmer over high heat. Reduce heat to a barest possible simmer (just a few bubbles breaking the surface) and cook, stirring occasionally, until the beef is tender but still somewhat firm and 1 1/2 to 2 cups thickened sauce surrounds the beef (about 2 hours).
Stir in the brown sugar, vinegar and more salt (if desired). Gently simmer 10 minutes more. (It might look like there is excess sauce.) Turn off the heat and let the chili stand for at least 30 minutes, during which time the beef will absorb about half of the skillet sauce. If mixture is too dry, stir in additional broth or water, or if sauce is not thick enough then let it simmer a bit more. Taste the sauce and adjust the salt, sugar or vinegar, if you like.
Serve in individual bowls and garnish with a dollop of sour cream and lime wedge on the side.