Preheat the oven to 375 degrees F. Lightly coat a 9x13-inch baking dish with nonstick spray.
In a large skillet, heat the olive oil over medium high. Once the oil is hot and shimmering, add the diced chicken. Sprinkle with salt and pepper. Sauté the chicken until golden on all sides and fully cooked through, about 4 minutes. Remove to a plate and set aside.
In a large stockpot of boiling, generously salted water, cook the pasta to al dente according to package instructions. Reserve 1 cup of the pasta cooking liquid, then drain. Return pasta to the stockpot.
While the pasta cooks, in a small bowl, whisk together the pesto and Greek yogurt until evenly combined.
Stir in the spinach, breaking it up as needed and stirring until evenly combined.
To the stockpot with the pasta, add the pesto/Greek yogurt mixture. Gently stir to coat the noodles in the pesto mixture. If the sauce is too thick, splash in some of the pasta water. It should be a little saucy.
Add the cooked chicken, cherry tomatoes, and red pepper flakes. Gently toss to combine.
Pour half of the pasta into the prepared baking dish. Sprinkle evenly with 1/2 cup mozzarella and half of the basil. Layer the remaining half of the pasta over the top, spreading it into an even layer. Sprinkle with the remaining 1/2 cup mozzarella.
Bake for 20-25 minutes until the cheese is melted and just barely starts to turn golden. Remove and sprinkle immediately with the Parmesan cheese and fresh basil. Serve warm.
TO MAKE AHEAD: Assemble the entire chicken pesto pasta in advance and refrigerate unbaked for up to 1 day. When ready to bake, let it come to room temperature, then bake as directed just before serving.
TO STORE: Place the baked chicken pesto pasta in an airtight storage container and store it in the refrigerator for up to 4 days.
TO FREEZE: You can freeze individual portions or large quantities of this chicken pesto pasta, either before or after baking. Place your desired amount into a freezer-safe storage container or ziptop bag and store in the freezer for up to 3 months.
TO REHEAT: Defrost chicken pesto pasta in the refrigerator overnight. Then, either bake at 375 degrees F until hot in the center or microwave gently until warm.
TO MAKE GLUTEN FREE: Swap gluten free pasta noodles.