In a deep skillet, bring the milk, pesto, and salt to a gentle simmer. Add the gnocchi, breaking it apart with your fingers if any are stuck together.
Cook for about 5 minutes, stirring occasionally until the gnocchi are tender and the sauce becomes creamy. Stir in the Parmesan cheese until it melts.
Taste the sauce and add salt (we usually add 1/8 teaspoon) and if it needs it, the final 1 tablespoon pesto. Add a bit of lemon zest if you have it, which freshens the flavors even more. The amounts will vary since all pesto brands are differently salted and flavored. Serve immediately. The sauce becomes very thick as it sits, so if necessary, add a splash of milk and reheat, stirring until creamy. Store leftovers refrigerated for 3 days and reheat in the same way. (You can also serve this as a main dish like pasta: just add some vegetables and additional protein.)