Bring a large pot of water with the tablespoon salt to a boil. Melt butter with the crushed garlic in a wide skillet over low heat for 2 minutes without browning.
Add cream, sauce salt and white pepper and simmer very gently for 3 minutes. Remove and discard the garlic clove.
Cook fresh fettuccine according to its package time until just tender, reserving 1 cup pasta water before draining. Add pasta to the warm skillet with 1/4 cup reserved water.
Remove from heat and toss while adding Parmesan gradually until smooth, using more pasta water a tablespoon at a time if needed. Divide among warm bowls and serve immediately.