To make sauce: Tear chiles into long strips. Discard stems and seeds. Heat a large heavy skillet over medium heat. Toast chiles, pressing down with a wide spatula and turning once or twice, until they give off a roasted aroma, about 2 minutes. Transfer the chiles to a bowl. Add boiling water to cover. Let sit until softened, about 20 minutes. Drain.
Combine the softened chiles, broth, garlic, tomato paste, sugar and oregano in a blender. Blend until very smooth. (If the mixture is too thick, add just enough water to give the sauce a smooth consistency.)
Heat oil in a medium heavy-bottomed saucepan over low heat. Add chile puree and cook, stirring constantly, until sauce has darkened and thickened slightly, about 5 minutes. Remove from heat and strain through a fine sieve. Adjust seasonings with salt and more sugar, if desired.
To grill venison: Preheat a grill or heat a cast-iron grill pan or skillet over high heat. Rub venison chops on both sides with garlic and brush lightly with oil. Season with salt and pepper.
Grill or sear the chops until browned on the outside but still pink in the center, 1 to 2 minutes per side.
Make Ahead Tip: Cover and refrigerate the sauce (Steps 1-3) up to 1 week ahead. Bring to a simmer before serving.
1 vegetable, 6 lean meat