1. MAKE THE CHILI: In a heatproof bowl, cover the ancho with 2 cups of the boiling water. Put the porcini in a small bowl and cover with the remaining 1 cup of boiling water. Let both soak until softened, about 20 minutes. Remove the ancho from the soaking liquid and discard the stem and seeds. Finely chop the chile; reserve the soaking liquid. Rub the porcini to remove any grit and finely chop them; reserve the soaking liquid.
2. In a large enameled cast-iron casserole, heat 2 tablespoons of the olive oil until almost smoking. Add half of the ground buffalo in an even layer and season with salt and pepper. Sear the meat over high heat for 3 minutes without stirring; leave the meat in large pieces. Gently stir the meat and continue to brown it for 2 minutes more. Using a slotted spoon, transfer the meat to a plate; brown the remaining meat.
3. Heat the remaining 1 tablespoon of olive oil in the casserole. Add the onions and cook over low heat until softened, about 10 minutes. Add the garlic and cook, stirring, until fragrant, about 2 minutes. Add the chile powder and cook, stirring, for 4 minutes. Add the tomato paste mixture, chicken stock, ancho and porcini. Add the ancho and porcini soaking liquids, stopping when you reach any grit at the bottom. Season with a large pinch of salt and simmer over moderately low heat until the sauce reduces by a third, about 10 minutes. Gently stir in the meat and any accumulated juices and season with salt and pepper. Rewarm just before serving.
4. MAKE THE FRY BREAD: Sift together the flour, cornmeal, baking powder and salt. With a wooden spoon, stir in the water until just blended.
5. In a 2-quart saucepan, heat 2 inches of oil to 350°. Brush a 1/2-cup measuring cup with oil and have 2 wooden spoons ready. Scoop a 1/2-cup portion of the batter into the hot oil, then use the 2 spoons to carefully spread the batter out to a 6- to 7-inch disk. Fry the bread until golden brown and cooked through, about 2 minutes per side; lower the heat if the oil becomes too hot. Drain on paper towels and transfer to a dinner plate. Top with the chili and serve at once. Repeat to make 7 more fry breads and serve with the remaining chili.
Make Ahead: The chili can be refrigerated for up to 3 days; reheat gently. The fry bread batter can stand at room temperature for up to 30 minutes.