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To a large pot, add the oil, garlic, carrot, celery, onion, and ¼ teaspoon of the salt. Turn the heat to medium-low and cook, stirring occasionally, until the vegetables are soft but not browned, about 20 minutes.
Add the tomato sauce, potatoes, and remaining salt and cook, stirring occasionally, until the potatoes begin to soften, about 20 minutes.
Pour in 5 cups of the stock, turn the heat to medium, and cover. Simmer until the potatoes are tender when pierced with a fork, about 20 minutes.
Uncover the pot, turn the heat to high, and add the pasta and enough of the remaining broth to cover by 1 inch. Cook—adding more broth ¼ cup at a time as needed to keep the pasta submerged—until al dente, 8–12 minutes. The texture should be like thick, chunky soup.
Remove from the heat, stir in the provolone, and serve.