Bring a pot of well-salted water to a boil on your camp stove. Cook pasta to al dente; drain and set aside (reserve ½ cup pasta water).
In a large 12" deep skillet or Dutch oven, heat 1 tbsp olive oil over medium-high. Season chicken with a pinch of salt and pepper, then sauté until lightly browned and cooked through, 5–7 minutes. Transfer to a bowl.
In the same pan, add remaining 1 tbsp olive oil and the butter. Sauté mushrooms with a pinch of salt until golden and their moisture cooks off, 4–5 minutes.
Stir in garlic and Italian seasoning; cook 30 seconds until fragrant.
Pour in chicken broth, scraping up browned bits. Stir in cream and bring to a gentle simmer, 2–3 minutes.
Return chicken (and any juices) to the pan. Add Parmesan and stir until the sauce is smooth. If too thick, loosen with a splash of pasta water.
Add drained pasta and spinach; toss until the spinach wilts and everything is coated and creamy, 1–2 minutes.
Season to taste with salt, pepper. Serve hot with extra Parmesan.