Preheat oven to 400°F. Drain spinach well, and press between paper towels to remove excess moisture.

Heat 2 tablespoons of the butter in a medium skillet over medium. Add shallot and garlic, and cook, stirring often, until softened, about 2 minutes. Add flour; cook, whisking constantly, until lightly browned, about 1 minute. Whisk in cream until smooth. Add spinach, and cook, stirring occasionally, until mixture is very thick, 2 minutes. Stir in Parmesan, 1 teaspoon of the salt, and 1/2 teaspoon of the pepper. Remove skillet from heat, and cool 10 minutes.

Toss together fresh breadcrumbs, chopped parsley, chopped thyme, 1/2 teaspoon of the salt, 1/4 teaspoon of the pepper, and remaining 4 tablespoons butter in a medium bowl. 

Sprinkle mushroom cavities evenly with remaining 1/2 teaspoon salt and 1/4 teaspoon pepper. Spoon creamed spinach filling into cavities. Top evenly with breadcrumb mixture, pressing gently to adhere. Arrange stuffed mushrooms on an aluminum foil-lined baking sheet. Bake in preheated oven until topping is golden brown and mushrooms are tender, 22 to 25 minutes. Serve stuffed mushrooms immediately.