Preheat oven to 375 degrees Fahrenheit.
Place pate brisee dough between two pieces of parchment paper and roll into a circle about 16-inches in diameter and ¼-inch thick.
In a small bowl, combine the mascarpone, 1 tablespoon chives, ¼ teaspoon salt and dash of pepper. Remove the top piece of parchment paper from the pate brisee and generously spread on the mascarpone, leaving about a 1 ½- to 2-inch border to fold the dough up and over the toppings.
Next, sprinkle on the fontina cheese, then top with the garlic, shallots and finally, the carrots. Drizzle olive oil over top of the carrots and season lightly with salt and pepper.
Working your way around the perimeter of the galette, fold up the edges of the dough, allowing them to naturally pleat as you go around in a circle. If any pieces crack, simply pinch them back together.
Keeping the galette on the bottom layer of parchment paper you originally rolled it out on, transfer to a large rimmed baking sheet. Using a pastry brush, brush the beaten egg over the exposed edges of the galette.
Place the galette in the oven for 40-45 minutes, or until the crust has turned a lovely golden brown. Remove and let cool slightly. Sprinkle with additional chives and fresh oregano before serving.