Preheat the oven to 400°F (200°C). Choose a casserole dish large enough to hold the 4 rolled tortillas and set it aside.
Heat a skillet over medium-high heat and add the 1 tablespoon olive oil. Add the minced garlic and cook until fragrant, about 1 minute.
Add the shredded cooked chicken to the skillet and cook just until warmed through, 1–2 minutes. Remove from heat.
In a bowl, stir together the 2 tablespoons sour cream, 1 cup salsa verde, and about half of the chopped cilantro. Season the sauce with salt and pepper to taste.
Add about half of the sauce mixture to the warmed shredded chicken and stir to combine; reserve the remaining sauce for topping.
If desired, warm the tortillas briefly so they are pliable (microwave a few seconds or heat in a dry skillet). Divide the chicken filling evenly among the 4 tortillas, spooning the filling down the center of each. Roll each tortilla tightly and place them seam-side down in the prepared casserole dish.
Spread the reserved sauce evenly over the rolled tortillas, then sprinkle the 1/2 cup shredded Mexican or cheddar cheese over the top.
Cover the casserole with foil and bake in the preheated oven for 15 minutes. Remove the foil and bake uncovered for an additional 10 minutes, until the cheese is melted and lightly golden.
Remove from the oven, let rest briefly, then garnish with the remaining chopped cilantro and the chopped red onion. Season with additional salt and pepper if desired and serve.