Poke holes into each sweet potato using a fork and place in a microwave-safe dish. Add enough water to fill dish about 1/2-inch deep. Cook sweet potatoes in the microwave until tender, about 12 minutes. Transfer sweet potatoes to a work surface until cool enough to handle.
Heat oil in a skillet over medium heat; cook and stir onion and rosemary until onions are softened, 12 to 15 minutes. Remove and discard rosemary sprigs. Stir brown sugar into onions; cook and stir until onions are browned and caramelized, 3 to 4 minutes. Stir mustard into caramelized onions and remove skillet from heat.
Slice sweet potatoes into 1/4- to 1/2-inch rounds.
Press each pie crust into a 9-inch pie dish. Line both crusts with caramelized onions and top with 1/2 of the sweet potatoes. Sprinkle 1/2 of the goat cheese and 1/2 of the Monterey Jack cheese over sweet potato layer. Spread the remaining 1/2 of the sweet potatoes over the cheese layer and top with remaining goat cheese and Monterey Jack cheese.
Beat eggs, milk, and Italian seasoning together in a bowl and pour over each filled pie crust until evenly filling out all the spaces around the sweet potatoes and cheeses.
Bake in the preheated oven until quiches are set in the middle, about 40 minutes.