In a medium bowl, combine the shredded chicken, Monterey Jack cheese, and sautéed onions and peppers. Mix until evenly combined and season lightly with salt and pepper if needed.

Warm the flour tortillas briefly in the microwave so they are soft and flexible. Place about one-third cup of filling in the center of each tortilla. Fold in the sides, then roll tightly like a burrito, placing the chimichangas seam-side down.

Heat oil in a deep skillet to about 350°F (175–180°C). Fry the chimichangas in batches, seam-side down first, turning as needed, until golden brown and crispy on all sides. Transfer to a wire rack or paper-towel-lined plate to drain.

In a saucepan over medium heat, melt the butter. Whisk in the flour and cook for 1–2 minutes until bubbly and lightly golden. Slowly whisk in the half-and-half, stirring constantly, until thick and smooth. Reduce heat to low, add the cheese, cumin, salt, and pepper, and stir until the sauce is creamy and silky.

Place the hot chimichangas on plates, spoon warm white cheese sauce generously over the top, and garnish with your favorite toppings. Serve immediately.