Heat the oven to 400º F. Line a baking sheet with parchment paper. Onto the sheet, place the tomatoes, garlic halves, and three stalks of par-boiled corn (make sure that this corn has already cooked in boiling water for 10-15 minutes).
Drizzle 3 tbsp olive oil on top the tomatoes, garlic, and corn. Season with 1 tsp salt and 1/2 tsp pepper. Coat everything generously in the oil, ensuring that the sliced halves of the garlic are well-coated, plus the corn on all sides. Place the garlic halves slice-side down on the sheet.
Roast vegetables for approximately 25-30 minutes, or until the garlic is golden brown and cooked-through (tomatoes and corn should also be softened, cooked, and have a golden hue on the skin). Set aside and let cool.
In a large dutch oven or soup pot, drizzle 2-3 tbsp olive oil. Add the diced carrots, minced onion, and celery. Season with a pinch of salt and pepper. Stir throughly and cook for 5-8 minutes, or until vegetables are softening and onion is getting translucent.
Once the carrot mixture is cooked and getting soft, add the roasted tomatoes and garlic to the pot. The cloves can be squeezed out of the head right into the pan (they should pop out and be soft and golden-brown). Stir thoroughly, breaking up the garlic cloves.
Add 1 quart of chicken stock to the pot, scraping the bottom of the pan with a wooden spoon to release the fond (the brown, flavorful bits that stick to the bottom). Let simmer for 10 minutes so that the vegetables get soft.
Use an immersion blender to completely puree the vegetable-stock mixture in the soup pot. If you do not have one, you can remove the mixture from the heat, let cool for 15 minutes, and puree in a blender. The pureed soup base should be thick and smooth. Return the mixture to the pot.
To the pureed mixture, add the remaining chicken stock. Then add the diced potatoes. Cover the pot with a lid and let the potatoes simmer on low-medium heat (not boiling, but simmering), for approximately 1 hour or until a fork can easily pierce a potato cube.