Heat the oil in a medium-large (5- to 6-quart) saucepan over medium-high. Sprinkle both sides of the chicken breasts first with salt, then with 1 tablespoon of the ground ancho chile. Lay them into the hot oil in a single layer. When browned on one side, 2 to 3 minutes, flip them over and brown the other side (another 2 to 3 minutes). Remove to a plate to cool, leaving behind as much oil as possible.
Add the onions and rice to the pan. Stir for several minutes until the rice turns from translucent to opaque. Add the garlic and remaining 1 ½ tablespoons of the ground ancho chile, cook 1 minute, then add the broth and 1 teaspoon salt (a little less if using salty broth). Stir well. When the mixture comes to a boil, reduce the heat to medium-low and cover the pot. Cook for 10 minutes.
Cut the chicken breast into 1-inch cubes. Uncover the pan and add the chicken and rinsed beans. Re-cover and cook 12 minutes longer.
Uncover, sprinkle on the green onions (or herbs) and test a kernel of rice: It should be have no more than a hint of chalkiness in the center; if it does, set over the heat for another 5 minutes or so. Otherwise, simply re-cover the pot and let stand off the heat for 5 to 10 minutes to finish the cooking with its own trapped steam. Fluff the rice mixture and serve with the salsa for each person to spoon on al gusto.