Peel and clean squash. Then grate using the large hole on a cheese grater. Set squash aside.
Set skillet over medium heat, add oil and swirl to ensure it coats the pan evenly.
Add onion and sauté at a low heat to soften, but do not brown the onion.
Add the garlic after the onion has softened, about 5-7 minutes. Cook for another minute.
Add grated squash, salt, and pepper.
Stir and cook the mixture until the squash has softened, 6-10 minutes. If the mixture becomes dry and wants to stick, add another tablespoon of oil as needed. The best way to check if the squash is ready is to taste it. If it's soft and edible, it's ready!
Remove from heat and add goat cheese, breaking apart with fingers, diced fontina, and cheddar cheese. Also, add the chives and thyme. Stir together to combine.
Place in a preheated 375°F oven until the sides are bubbling and the dip is hot in the middle, 15-20 minutes. Different sized baking dishes may need an adjusted baking time.
Remove from oven and sprinkle with more chives. Serve warm with baguette slices, chips, crackers, or vegetables.
If you do not have a cast-iron skillet, any oven-safe skillet will work. Alternatively, you can sauté onion and garlic, then mix with cheese and put in an oven-safe baking dish.