Cook the pasta: Bring a large pot of salted water to a boil. Cook pasta until al dente.
In the last 2 minutes, add the broccoli to the pot. Drain and set aside, reserving 1/2 cup of pasta water.
Season the chicken: Pat chicken dry and season with 1/2 teaspoon salt and 1/4 teaspoon black pepper.
Sear the chicken: Heat 1 tablespoon olive oil in a large skillet over medium-high. Add chicken and cook 5–7 minutes, stirring occasionally, until browned and cooked through.
Transfer to a plate.
Sauté the garlic: Reduce heat to medium. Add remaining 1 tablespoon olive oil to the same skillet. Add garlic and red pepper flakes.
Cook 30–45 seconds until fragrant, not browned.
Build the sauce base: Sprinkle in flour and stir for 30 seconds. Slowly whisk in chicken broth, then milk, stirring to smooth out any lumps. Simmer 2–3 minutes until slightly thickened.
Add the creamy elements: Take the pan off the heat.
Whisk in Greek yogurt, Parmesan, and lemon juice until smooth. If the sauce seems too thick, whisk in a splash of reserved pasta water. Season with salt and pepper.
Combine everything: Add the cooked chicken, pasta, and broccoli to the skillet.
Toss gently over low heat for 1–2 minutes until coated and warmed through. Add more pasta water as needed for a silky sauce.
Finish and serve: Taste and adjust seasoning. Top with parsley and a bit more Parmesan.
Serve right away.