Sauté chorizo in an 8-inch nonstick omelet pan or skillet with sloped sides 7 to 10 minutes or until browned. Remove from skillet, and drain on paper towels. Wipe skillet clean.
Melt 1 Tbsp. butter in skillet, and sauté green onions, bell pepper, and jalapeño peppers over medium-high heat 3 to 5 minutes or until tender. Transfer to a bowl; stir in chorizo. Wipe skillet clean.
Whisk together cornmeal mix, buttermilk, milk, all-purpose flour, and 1 large egg.
Coat skillet with cooking spray; melt 1 Tbsp. butter in skillet over medium-high heat, rotating pan to coat bottom evenly. Pour about 1/3 cup cornmeal mixture into skillet. Tilt pan so uncooked portion flows around to coat bottom of pan, cooking until almost set, bubbles form, and edges are dry (about 1 1/2 minutes). Gently flip with a spatula.
Sprinkle 1 side of omelet with about 1/2 cup onion mixture and about 3 Tbsp. cheese. Fold omelet in half; cook 30 seconds or until cheese is melted. Transfer to a serving plate; keep warm. Repeat procedure 4 times with remaining butter, cornmeal mixture, onion mixture, and cheese. Serve immediately.