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 to the skillet, season with salt and pepper, and cook until the onion is tender, about 5 minutes. Remove the skillet from heat.
In a mixing bowl, beat together the cream cheese and Greek yogurt (or sour cream) until smooth. Stir in the chili powder, smoked paprika, cumin, and a pinch of salt and pepper.
Add the cooked onions and peppers to the cream cheese mixture, then stir in 1 1/2 cups of the salsa verde and the shredded chicken until evenly combined.
Return the skillet (if removed) to the work surface. Spoon a portion of the chicken mixture into the skillet to make the first layer, then sprinkle with cheese and add a layer of tortilla chips.
Repeat layering with chicken mixture, cheese, and chips to create about three layers, ending with the chicken mixture on top and reserving a handful of cheese for the final topping.
Pour the remaining salsa verde over the layered skillet and finish with the reserved cheese.
Bake in the preheated oven for 25–30 minutes, until the cheese is melted and bubbling.
Remove from the oven and serve immediately, topped with avocado, cilantro, green onion, and/or basil as desired.