In a medium saucepan, heat the milk, butter, mustard, and salt over medium heat until the butter has fully melted and the milk is hot. Using a wooden spoon or silicone spatula, stir in the flour all at once and continue mixing until the dough comes together and pulls away from the sides of the pot. Reduce the heat to low and cook the dough for another 1 to 2 minutes to dry it out slightly.

Transfer the dough to a large mixing bowl and let it cool for 5 to 10 minutes. Stir in the grated cheese, then add the eggs one at a time, beating well after each addition with a wooden spoon, electric hand mixer, or stand mixer fitted with the paddle attachment. The dough should be thick and glossy, similar to a smooth pâte à choux. Spoon the dough into a large piping bag or sturdy zip-top bag and snip off the tip to create a 3/4-inch-wide opening.

Bring a large pot of salted water to a boil. Working in batches, hold the piping bag over the pot and use kitchen scissors to snip off 1-inch pieces of dough, letting them fall directly into the simmering water. Cook for about 3 minutes, or until the gnocchi float and feel firm. Remove with a slotted spoon and set aside.

In a large sauté pan or skillet, melt 1 tablespoon of butter over medium-high heat. Add the diced zucchini and mushrooms and sauté until browned and softened, about 5 to 7 minutes. Add the garlic and cook for 30 seconds more. Deglaze the pan with the white wine, scraping up any browned bits, and let the wine reduce by half.

Stir in the heavy cream, basil leaves, and grated Parmesan. Let the sauce simmer gently until slightly thickened, about 3 to 4 minutes. Season with salt and freshly ground pepper to taste.

Add the cooked gnocchi to the sauce and stir gently to coat. Simmer for another minute until everything is heated through. Serve hot, garnished with additional Parmesan if desired.