Working in batches, add the mushrooms slices to the skillet and brown them on both sides. Season the mushrooms with salt and set them aside.
Place the asparagus in the skillet and a generous pinch of salt. Sauté until just tender. Smash and peel two cloves of the garlic and set them aside.
Return the mushrooms to the skillet and add a tablespoon of the butter, the smashed garlic, and the thyme sprigs.
Deglaze the pan with the cognac and simmer until it reduces by half, this should only take a few moments. Add the Dijon mustard followed by the cream.
Place the peppercorns in a mortar and pestle and coarsely grind them, set them aside. Drain the pasta and place it in the sauce over low heat. Return the vegetables to the skillet and toss to coat.
Divide the pasta across four bowls and garnish with the cracked peppercorns, a sprinkling of arugula, and Grana Padano. Serve immediately.