Preheat the oven to 375°F (190°C).
Heat the butter or oil in an 11–12-inch oven-safe skillet over medium-high heat.
Add the chopped onion and poblano peppers, season with salt and pepper, and cook until the onion is tender, about 5 minutes. Remove the skillet from the heat.
In a mixing bowl, beat together the cream cheese and Greek yogurt (or sour cream) until smooth. Stir in the cooked onions and peppers, chili powder, smoked paprika, cumin, and additional salt and pepper to taste.
Mix in 1 1/2 cups of the salsa verde and the shredded chicken until evenly combined.
Return the skillet to the stove if needed. Spoon a portion of the chicken mixture into the skillet to form the first layer, then sprinkle some shredded cheese and top with a layer of tortilla chips.
Repeat layering: chicken mixture, cheese, and chips to create a total of three layers, finishing with the chicken layer and reserving a handful of cheese for the top.
Pour the remaining salsa verde evenly over the layered skillet, then sprinkle the reserved cheese on top.
Bake in the preheated oven for 25–30 minutes, until the cheese is melted and bubbly.
Remove from the oven and serve immediately, topped with avocado, cilantro, green onion, and/or basil as desired.