Dirty Martini Potato Crust Quiche

Name: Dirty Martini Potato Crust Quiche
Ingredients: 1/4 cup (2 ounces) gin; 2 tablespoons (1 ounce) dry vermouth; 2 tablespoons drained jarred diced pimento; 1 cup pitted Castelvetrano olives, drained and quartered lengthwise, plus 3 tablespoons brine from jar, divided; 6 tablespoons unsalted butter, melted and divided; 1 (30-ounce) package frozen shredded hash browns (such as Ore-Ida), thawed and squeezed dry with a clean kitchen towel; 3 ounces Parmesan cheese, grated on smallest holes of box grater (about 3/4 cup), divided; 2 teaspoons plus a pinch of kosher salt, divided; 3/4 teaspoon plus a pinch of black pepper, divided; 8 large eggs; 1 cup heavy whipping cream; 2 ounces Fontina cheese, shredded (about 1/2 cup); 2 1/2 ounces Gorgonzola Dolce, crumbled (about 1/2 cup plus 2 tablespoons), divided; 1 teaspoon Dijon mustard; 1 tablespoon olive oil; 4 cups packed spring mix salad greens