Preheat the oven to 375°F (190°C).
Place the diced potatoes in a large pot and cover with water. Bring to a boil and cook until tender, about 10-12 minutes. Drain and set aside.
In a medium saucepan, melt the butter over medium heat. Add the chopped onion and cook until softened, about 3-4 minutes.
Add the minced garlic to the saucepan and cook for 1 minute until fragrant.
Whisk in the flour and cook for 1-2 minutes to form a roux, stirring constantly.
Gradually whisk in the whole milk and continue to cook, stirring frequently, until the sauce thickens, about 3-5 minutes.
Remove the sauce from heat and stir in the sour cream, salt, black pepper, smoked paprika, and dried thyme until smooth.
In a large mixing bowl, combine the cooked potatoes, shredded chicken, thawed peas, shredded cheddar cheese, shredded mozzarella cheese, and the prepared sauce. Mix gently to combine.
Transfer the mixture to a greased 9×13 inch baking dish and spread evenly.
Sprinkle the grated Parmesan cheese evenly over the top.
Bake in the preheated oven for 30-35 minutes, or until the casserole is bubbly and the top is golden brown.
Remove from the oven and let it rest for 5 minutes before serving.