Poke holes into each sweet potato using a fork and place in a microwave-safe dish. Add enough water to fill dish about 1cm 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 pan 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 caramelised, 3 to 4 minutes. Stir mustard into caramelised onions and remove pan from heat.
Slice sweet potatoes into 1cm rounds.
Line 2 quiche tins with pastry bases; spread caramelised onions on top and then 1/2 of the sweet potatoes. Sprinkle 1/2 of the goat cheese and 1/2 of the grated 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 grated cheese.
Beat eggs, milk and Italian herbs together in a bowl and pour over each filled pastry base until evenly filled around the sweet potatoes. If you like a smooth top, press down on each quiche gently until level before baking.
Bake in the preheated oven until quiches are set in the middle, about 40 minutes.