Bring a large pot of salted water to a boil. Cook pasta 1 minute less than package directions for al dente. Reserve 1/2 cup pasta water before draining. Drain and set aside.

While pasta cooks, pat chicken dry with paper towels. Season both sides with 1/2 teaspoon salt and 1/4 teaspoon black pepper.

Heat olive oil in a large deep skillet over medium-high heat until shimmering. Add chicken breasts and cook 5-6 minutes per side, until golden brown and internal temperature reaches 165°F (74°C) on an instant-read thermometer. Transfer to a cutting board and let rest 5 minutes, then slice crosswise into 1/2-inch strips.

Reduce heat to medium. Add garlic and red pepper flakes to the skillet. Cook 30 seconds until fragrant, stirring constantly to prevent burning.

Pour in chicken broth and lemon juice. Scrape up any browned bits from the bottom of the pan with a wooden spoon. Let simmer 1 minute.

Whisk in milk and lemon zest. Bring to a gentle simmer, then stir in Parmesan until melted and smooth. The sauce will be thin at this point.

Add spinach to the skillet. Toss with tongs for 1-2 minutes until just wilted.

Add drained pasta and sliced chicken to the skillet. Toss everything together. If the sauce seems too thick, add reserved pasta water a tablespoon at a time until it coats the pasta nicely. Taste and adjust salt and pepper.

Serve immediately with extra Parmesan and a pinch of lemon zest on top.