In a large pot of salted boiling water, cook pasta according to package directions until al dente. Drain, reserving 1 cup pasta water.
Meanwhile, in a large skillet over medium-high heat, heat oil. Season chicken with salt and pepper and cook until golden and no longer pink inside, about 8 minutes per side. Let rest 10 minutes, then thinly slice.
Meanwhile, in same skillet, cook bacon over medium heat until crispy, 8 minutes. Drain on a paper towel–lined plate, then chop. Pour off half of fat from skillet.
Add garlic, tomatoes, and spinach to skillet and cook over medium heat until fragrant and slightly wilted, 2 minutes. Season with salt and pepper, then add heavy cream, Parmesan, and 1/2 cup of reserved pasta water. Simmer 5 minutes.
Add cooked pasta and toss until fully coated, then add chicken and bacon and toss until combined. (For a looser sauce, stir in more reserved pasta water.)