Heat olive oil in heavy medium saucepan over medium-low heat. Add onion and chilies; cook until onion is translucent, stirring occasionally, about 8 minutes. Add chili powder and cumin; stir 30 seconds. Add beans and lime juice. Cook until heat through, stirring and mashing beans slightly with spoon, about 4 minutes. (Can be prepared 1 day ahead. Cover and chill.)
Heat oil in heavy large skillet over high heat. Add chicken and sprinkle with salt and pepper. Stir until almost cooked through, about 3 minutes. Add chili powder and cumin and stir until cooked through, about 30 seconds. Remove from heat.
Combine first 3 ingredients in large bowl.
Add oil to depth of 1/4 inch to heavy medium skillet. Heat over medium-high heat until just beginning to smoke. Add tortilla and cook until crisp, abut 30 seconds per side. Drain on paper towels. Repeat with remaining tortillas.
Add 1/4 cup olive oil to salad; toss Season with salt and pepper. Add 1 tablespoon lime juice; toss. Rewarm beans over medium-low heat, stirring and thinning slightly with water. Place 1 tortilla on each plate. Spread with beans. Sprinkle with cheese. Top with salad, then chicken, salsa and cilantro.
