Preheat the oven to 400°F (200°C). Lightly grease a casserole dish.
Heat the olive oil in a skillet over medium-high heat. Add the minced garlic and cook until fragrant, about 1 minute.
Add the shredded cooked chicken to the skillet and warm through, stirring for 1–2 minutes; remove from heat.
In a mixing bowl, combine the sour cream, 1 cup salsa verde, and half of the chopped cilantro. Season with salt and pepper and stir until smooth.
Mix half of the sauce into the warm shredded chicken to make the filling.
Divide the chicken filling among the tortillas, roll each tortilla tightly, and place seam-side down in the prepared casserole dish.
Pour the remaining sauce over the rolled tortillas and sprinkle evenly with the shredded Mexican or cheddar cheese.
Cover the casserole with aluminum foil and bake for 15 minutes. Remove the foil and bake an additional 10 minutes, until the cheese is melted and lightly golden.
Remove from the oven and garnish with the remaining chopped cilantro and the chopped red onion before serving.