Preheat your oven to 375°F. While the oven heats, season the chicken breasts with salt and pepper on both sides. Heat a grill pan or skillet over medium heat, drizzle it with a little olive oil, and cook the chicken until it’s golden and fully cooked through. Remove the chicken and shred it with two forks. Set aside to cool.
Combine ricotta, cottage cheese, mozzarella, 1 cup of Parmesan, eggs, parsley, salt, and pepper in a large bowl. Add the shredded chicken and mix everything until incorporated. Add a splash of heavy cream if the mixture feels too thick but don’t overdo it.
Melt the butter in a large skillet over medium heat. Sprinkle in the flour and whisk it together, letting it cook until golden brown. Slowly pour in the milk and heavy cream, whisking constantly to avoid lumps. Once the sauce thickens, stir in minced garlic, 1 cup Parmesan, basil, parsley, and salt. Taste the sauce to adjust seasonings if needed.
Spread 1 cup of the Alfredo sauce onto the bottom of a 9x13-inch baking dish. Take the cooked pasta shells and spoon in a generous amount of the chicken and cheese filling. Place each stuffed shell face down into the sauce base. Pour the remaining Alfredo sauce over the shells, ensuring they’re fully covered. Sprinkle more mozzarella and Parmesan on top.
Bake the dish in the preheated oven for 25 minutes or until the top is golden and bubbly. Allow it to cool for a few minutes before garnishing with extra parsley and basil.