Tart Shell

• 1 1/2 cups all-purpose flour
• 1 stick cold unsalted butter (4 ounces), cut into small pieces
• Pinch of kosher salt
• 1/4 cup ice water

Filling

• 2 tablespoons unsalted butter
• 1/2 pound white mushrooms, sliced 1/4-inch thick
• 3/4 teaspoon kosher salt, plus more for seasoning the mushrooms
• 1/4 teaspoon freshly ground black pepper, plus more for seasoning the mushrooms
• 4 large eggs
• 3/4 cup heavy cream
• 1/2 cup milk
• 3 ounces ham, sliced 1/4-inch thick and cut into 1/4-inch matchsticks
• 1 cup shredded Gruyère or other Swiss-type cheese (3 ounces)