In the bowl of your slow cooker, layer the onion and then the chicken. Sprinkle the chicken with garlic powder, paprika, chili powder, and season with salt and pepper. Pour over the salsa verde and 1/4 cup water. Cover and cook on low for 3-4 hours or on high for 1-2 hours.

Crank the heat on the slow cooker to high. Stir in the milk, cream cheese, and cheddar cheese. Add the broccoli. Cover and cook for 15 minutes, or until the sauce is creamy.

Melt together the butter, garlic, and thyme leaves in a small skillet set over medium heat. Cook until the butter is browning and the thyme is crisp, 4 minutes.

Serve the chicken, broccoli, and sauce topped with brown butter and fresh basil.