Season Chicken: Sprinkle the chicken cutlets on both sides with about half the salt and half the pepper.
Coat with Flour: Add the flour to a shallow dish. Press each cutlet into the flour to lightly coat both sides, shaking off any excess.
Heat the Pan: Set a large skillet over medium-high heat. Add the olive oil and butter and let the butter melt and foam.
Cook the Chicken: Add the chicken cutlets and cook until golden and cooked through, about 3–4 minutes per side. Transfer to a plate.
Sauté the Mushrooms: Add the sliced mushrooms to the same skillet and cook, stirring, until softened and lightly browned, about 4–5 minutes.
Add Garlic and Thyme: Stir in the minced garlic and dried thyme and cook until fragrant, about 30 seconds.
Season the Mushrooms: Sprinkle in the remaining salt and pepper, then stir to combine.
Add Pasta and Liquid: Add the linguine to the skillet. Pour in the chicken stock and half-and-half and press the pasta down so it’s as submerged as possible.
Simmer Until Tender: Bring to a boil, then reduce to a steady simmer. Cover and cook, uncovering to toss and stir every couple minutes, until the linguine is tender and the sauce looks creamy, about 10–12 minutes. If the pan gets dry before the pasta is done, add a splash more stock.
Melt in the Asiago: Sprinkle in the shredded Asiago and toss until melted and the sauce thickens slightly.
Wilt the Spinach: Add the baby spinach and stir until wilted, about 1 minute.
Finish and Serve: Slice the chicken and return it to the skillet (or serve it on top). Taste and adjust salt and pepper if needed, then serve hot while it’s still gloriously creamy.