Preheat oven to 400 degrees F (200 degrees C).

Heat olive oil in a skillet over medium-high heat. Saute onion and garlic in hot oil until onion is soft and translucent, 5 to 7 minutes. Stir chicken, chiles, rosemary, and thyme into onion mixture. Cook and stir until chicken is heated through, about 5 minutes.

Stir cream of chicken soup and evaporated milk together in a saucepan over medium-high heat until mixture is warm, 3 to 5 minutes.

Spread green beans into the bottom of an 8-inch square baking dish and pour chicken mixture over the beans. Top chicken layer with cream of chicken soup mixture and spoon mashed potatoes over the top. Sprinkle Parmesan cheese and basil over the mashed potatoes.

Bake in the preheated oven until potatoes are lightly golden and casserole is heated through, about 20 minutes.