Preheat broiler. In a bowl whisk together eggs, milk, cilantro, and salt.
In an extra-large broiler-proof nonstick skillet heat oil over medium heat. Add chiles. Cook and stir 3 minutes. Scatter chicken and beans evenly in skillet.
Pour egg mixture over chicken and beans. Cook over medium heat. As mixture sets, run a spatula around edge of skillet, lifting egg mixture so uncooked portion flows underneath. Continue cooking and lifting edges until egg mixture is almost set (surface will be moist). Sprinkle with cheese.
Place skillet under broiler 4 to 5 inches from heat. Broil 1 to 2 minutes or until top is just set and cheese is melted. Top with Chimichurri Sauce and fresh cilantro leaves. Cut into wedges.
In a food processor or blender combine the first seven ingredients (through crushed red pepper). With the processor running, slowly add olive oil through the opening in the lid until mixture is smooth.