Preheat oven to 375 degrees. Spray or oil an 11 x 7, 8 x 10 or other similar-sized baking dishes. Set aside.
Cut the potatoes into 1/2-inch cubes. Place in a saucepan. Fill with water to cover, and add approximately 1 tablespoon of salt. Bring to a boil and cook 4-5 minutes or until you can just pierce them with a knife. (They should still have some resistance; you don't want them falling apart.)
Drain in a colander and cool under cold running tap water. Drain again. Set aside.
Combine the butter and oil in the saucepan you used to cook the potatoes and melt the butter over medium-high heat.
Add the onion, reduce heat to low and cook the onion for 5-6 minutes or until softened, being careful not to brown it.
Add the flour and cook 1-2 minutes, whisking constantly.
Slowly add the half-and-half and bring to a simmer. Cook, whisking constantly until thickened. (Should be the consistency of heavy cream.)
Add the shredded cheese by the handful, whisking constantly after each addition melts and until the cheese is nicely melted and incorporated. Taste for seasoning and add salt and black pepper to taste.
Pour approximately 3/4 cup of the sauce to cover the bottom of the prepared baking dish.
Place approximately 1/3 of the potatoes in the prepared baking dish, and top with approximately 1/3 of the green chiles.
Drizzle with some of the sauce and repeat the layers two more times, finishing with the remaining sauce.
Cover loosely with aluminum foil. Bake for 30-35 minutes. Remove the foil and continue baking another 20-25 minutes or until bubbling and the top is nicely browned.
Let rest approximately 5 minutes then serve.