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.

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

Meanwhile, 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. Fold in cheese.

Spoon chorizo mixture in a line across the middle of a tortilla, then add a layer of scrambled eggs. Fold opposing edges of the tortilla to overlap the filling. Roll 1 of the opposing edges around the filling creating a burrito.