Preheat the oven to 375 degrees.
Season the chicken on both sides with salt and pepper.
Heat a grill pan or an iron skillet over medium heat, and drizzle it with olive oil.
Add the chicken to the pan and cook, flipping once, until cooked through at 165 degrees F internally, about 5 minutes per side.
Transfer the chicken to a plate and shred it with 2 forks, then allow it to cool.
In a medium bowl, combine the ricotta, cottage cheese, mozzarella, 1 cup of parmesan, eggs, 2 tablespoons of parsley, salt, pepper, and the shredded chicken, stirring to combine. If the mixture seems too thick, add 2 tablespoons of heavy cream.
In a large skillet over medium heat, melt the butter.
Add the flour to the melted butter, whisking and cooking until it becomes a golden brown, about 2 minutes.
Pour in the milk and 1 cup of heavy cream, whisking constantly and cooking until thickened, about 2-3 minutes.
Add the garlic, remaining parmesan cheese, salt, pepper, 1 tablespoon of parsley, and basil, stirring until combined.
Taste the mixture for seasoning.
Add 1 cup of the sauce to the bottom of a 9x13-inch baking dish.
Using a spoon, fill each of the jumbo shells generously with the chicken mixture.
Place the shells face down in the pan.
Pour the remaining sauce over the top of the shells.
Sprinkle the extra mozzarella and extra parmesan over the top.
Bake until bubbly and golden, about 25 minutes.
Serve.