Mix crushed crackers and melted butter together in a bowl; press into the bottom and up the sides of a 9-inch springform pan.
Bake in the preheated oven for 10 minutes. Remove crust from oven and brush egg white onto the crust, to seal. Bake 4 minutes more. Remove from oven and cool to room temperature.
Season chicken with salt and pepper.
Heat olive oil in a heavy large skillet over medium heat; cook and stir chicken just until brown, about 2 minutes per side. Transfer chicken to a plate.
Reduce heat under skillet to medium-low; add 2 tablespoons butter. Cook and stir shallot in the hot butter until beginning to soften, about 2 minutes. Add mushrooms, garlic, rosemary, and 1 pinch salt; cook and stir until mushrooms are tender and juices evaporate, about 8 minutes.
Whisk cream, eggs, 1/4 cup grated Parmesan cheese, 2 tablespoons parsley, 1/2 teaspoon salt, and 1/2 teaspoon pepper together in a bowl.
Place baked crust on a baking sheet; top with chicken and mushrooms. Pour egg mixture over top. Sprinkle remaining 2 tablespoons Parmesan cheese over the top.
Bake in the preheated oven until top is lightly golden and middle is set, 30 to 35 minutes. Let tart rest for 10 minutes. Remove outer ring. Top tart with 2 tablespoons remaining parsley.