Drain ricotta cheese in a mesh strainer set over a bowl in the refrigerator, 8 hours to overnight.

Preheat the oven to 450 degrees F (230 degrees C).

Season chicken with salt, pepper, and cayenne.

Heat 2 tablespoons oil in a pan over medium-high heat. Cook chicken breasts in the hot oil until browned and just barely cooked through, about 6 minutes per side. Turn off heat and splash in cold water, stirring to deglaze. Transfer chicken and pan drippings into a bowl to let cool, about 5 minutes. Dice or shred the chicken.

Combine drained ricotta cheese, salt, pepper, 8 ounces mozzarella cheese, 8 ounces Monterey Jack cheese, and 1 ounce Parmigiano-Reggiano cheese in a bowl. Mix with a spoon to evenly distribute. Add chicken, the pan drippings, and marinara sauce. Mix with a fork until just combined.

Combine bread crumbs and remaining olive oil in the bowl used for the chicken and mix to coat.

Transfer chicken mixture to an ungreased baking dish. Top with oiled crumbs, 2 tablespoons Parmesan, 2 tablespoons mozzarella, and 2 tablespoons Monterey Jack cheese.

Bake in the center of the preheated oven until browned and bubbly, about 20 minutes.

Let rest for 10 minutes. Garnish with parsley and serve with a spoon or spreading knife.