Bring a large pot of salted water to a boil, then add the chicken tortellini and cook according to package directions, about 5–8 minutes for fresh tortellini. Drain and set aside.
Heat 1 teaspoon olive oil in a large skillet over medium heat. Add the sliced mushrooms and sauté until soft, about 4–6 minutes. Remove mushrooms from the pan and set aside.
In the same skillet over medium heat, combine the half and half, crumbled feta, grated Parmesan, Italian seasoning, and freshly cracked pepper. Whisk continuously until the cheeses melt and the sauce thickens, about 7–10 minutes. Reduce heat if the sauce begins to simmer too vigorously to prevent scorching.
Remove the skillet from the heat and add the drained tortellini, sautéed mushrooms, and the baby spinach in handfuls. Toss gently until combined; the residual heat will wilt the spinach and coat the pasta in sauce.
Serve immediately, seasoning with additional cracked pepper if desired.