Inspect fresh morel mushrooms for dirt and debris, rinse quickly under cold running water, and dry immediately.

Bring 4 quarts of salted water to a vigorous boil and cook tagliatelle according to package instructions for 2-3 minutes until just al dente.

Heat 3 tablespoons of unsalted butter in a skillet over medium-low heat, add finely minced shallots and a pinch of salt, and sweat for 4-5 minutes until they become translucent.

Raise heat to medium, add cleaned and halved morel mushrooms, and cook for 6-8 minutes until golden and caramelized.

Add minced garlic and fresh thyme leaves to the skillet and sauté for 30 seconds until fragrant.

Pour in dry white wine, stir well, and let simmer for about 2 minutes until the skillet is almost dry.

Transfer al dente tagliatelle from boiling water into the skillet, mixing in some starchy pasta water.

Pour in heavy cream, season with kosher salt and freshly ground black pepper, and toss the pasta in the sauce.

Remove from heat, add cold butter cubes, and toss until melted and sauce becomes glossy, then stir in freshly grated Parmigiano-Reggiano.

Plate the pasta, garnish with extra Parmigiano-Reggiano and fresh thyme leaves for presentation.