Heat a large skillet over medium heat. Add chorizo, poblano pepper, onions, and red bell pepper. Cook until vegetables are tender, 5 to 7 minutes. Remove skillet from heat.

Cook potatoes in a microwave-safe bowl with a small amount of water until fork-tender, stirring halfway through, approximately 7 to 10 minutes; drain. Stir potatoes into chorizo mixture.

Whisk eggs, half-and-half, salt, and pepper together in a large bowl.

Heat butter in a skillet over medium heat. Add egg mixture and scramble until eggs are cooked, about 3 minutes. Remove eggs from heat and fold in cheese.