Heat olive oil in a large sauce pot over medium heat. Add celery and onions, cook until soft and the onion is translucent, about 5 minutes. Toss garlic in, cook about 30 seconds.
Turn the heat to high and dump thyme, potatoes, and broth into the pot, stirring to combine. Bring liquid to a boil, then lower to a simmer and cook until potatoes are fork-tender but not falling apart, about 10-12 minutes.
Stir in corn, coconut milk, and cheese until warmed, thickened, and the cheese melts. I prefer shredding a block of cheese to obtain the needed amount rather than buying pre-shredded - I find the former softens much more easily.
Serve with additional mozzarella, fresh parsley, and baguette, as desired.