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 poblanos to the skillet, season with salt and pepper, and cook, stirring occasionally, until the onion is softened, about 5 minutes; remove from heat.
In a mixing bowl, beat together the cream cheese and Greek yogurt (or sour cream) until smooth, then stir in the chili powder, smoked paprika, and cumin.
Fold the cooked onion and peppers into the cream cheese mixture, then stir in 1½ cups of the salsa verde and the shredded chicken until combined.
Return the skillet to medium heat if it has cooled slightly; spoon a layer of the chicken mixture into the bottom of the skillet.
Top with a layer of shredded cheese and a layer of tortilla chips. Repeat to create three layers, ending with the chicken mixture on top and reserving a handful of cheese.
Pour the remaining salsa verde over the top, sprinkle with the reserved cheese, and transfer the skillet to the oven.
Bake for 25–30 minutes, until the cheese is fully melted and bubbly.
Remove from the oven and serve immediately topped with sliced avocado, cilantro, green onion, and/or basil as desired.
