Heat the oven to 425°F. Toss the sliced mushrooms with 1 tablespoon olive oil and a pinch of salt and pepper on a rimmed sheet pan. Roast until deep golden on the edges, 12–15 minutes. Set aside.
Season the chicken with lemon zest, dried thyme, 1/2 teaspoon kosher salt, and 1/4 teaspoon black pepper. Let it stand at room temperature for 10 minutes while you start the pasta water (this quick rest seasons the meat throughout).
Bring a large pot of water to a rolling boil. Salt generously, add the penne, and cook until just shy of al dente (about 1 minute less than package). Reserve 3/4 cup pasta water, then drain.
Warm a large skillet over medium-high heat with the remaining 1 tablespoon olive oil. Sear the chicken in a single layer until lightly browned and just cooked through, 2–3 minutes per side. Transfer to a plate and tent loosely with foil.
Reduce the heat to medium. Add the butter and the chopped shallot to the same skillet; cook, stirring, until softened, 2 minutes. Stir in the garlic for 30 seconds, just until fragrant.
Pour in the white wine, scraping up any browned bits. Simmer briskly until reduced to a syrupy layer, 3–4 minutes. Add the chicken broth and heavy cream; bring to a gentle simmer and cook 4–5 minutes until slightly thickened.
Return the chicken and any accumulated juices to the pan along with the roasted mushrooms. Sprinkle in the Parmesan a small handful at a time, stirring until smooth. Season with the remaining salt and pepper to taste.
Add the drained pasta and the baby spinach to the skillet. Toss to coat, loosening with a splash or two of the reserved pasta water until the sauce is glossy and clings to the penne. Cover, remove from heat, and let stand 2 minutes to marry flavors.
Uncover, adjust seasoning, and finish with chopped parsley and extra Parmesan. Serve immediately.