Place quartered potatoes in a large pot and add enough cold water to cover by 1-inch. Bring to a boil then reduce the heat to medium-high. Cook for 8-10 minutes, or until the potatoes are fork-tender but not falling apart. Drain the potatoes and set aside.

Set a large skillet over medium heat, then add the olive oil and butter. When the butter is melted, add the onion. Cook for 2-3 minutes, until softened. Stir in the garlic and continue cooking for an additional 30 seconds.

Add the cooked potatoes to the skillet in a single layer and stir gently to combine with the onion mixture. Cook 3-4 minutes, to heat through.

Stir in the green chilies, salt, and black pepper. Adjust seasoning to taste.

Remove from heat and sprinkle the cheese over the potatoes. Cover and let stand for 5 minutes, or until the cheese is melted. Serve hot.