This is a knock-your-socks off kind of dessert, but it's really quite simple to make, especially if you make it in stages. First poach the quince in spiced cider. While the quinces are cooking, you can make the frangipane (almond paste). Both of these steps can also be done a day ahead of time, which makes last-minute assembly and baking a snap. This recipe works equally well for individual galettes or for one large tart. It's best served warm with a dollop of créme fraiche or a scoop of vanilla ice cream.
Combine the cider, 1/2 cup of sugar, lemon zest and juice, cinnamon, allspice, and nutmeg in a medium bowl and whisk to blend. Working with one quince at a time, peel, core, and cut it into quarters. Drop the pieces into the cider mixture as you go so the fruit doesn’t discolor. Repeat with remaining quinces.
Position a rack in the middle of the oven and preheat to 325 degrees F.
Arrange the quince quarters in a baking dish big enough to accommodate them in a single layer. Pour the cider mixture over the fruit. Cover the pan tightly with a sheet of aluminum foil. Bake until the fruit is just tender when pierced with the tip of a knife, about 2 hours. Remove the pan from the oven. Transfer the quince with a slotted spoon to a platter and let cool. When they’re cool enough to handle, cut the quinces into 1/4-inch-thick slices and set aside.
Place the poaching liquid in a saucepan and bring to a boil over medium-high heat. Boil until the syrup reduces and thickens slightly, 20 to 25 minutes. The syrup and quince quarters can be refrigerated (separately), covered, for up to 2 days.
Meanwhile, place the almonds and the remaining 1/2 cup of sugar in the bowl of a small food processor. Run the machine until the mixture is a fine, even powder. Add the butter and process until the mixture is smooth. Add the eggs and egg whites, and run the machine until the mixture forms a smooth paste. Add the almond extract and process briefly to blend. If you’re not using it immediately, cover and refrigerate the frangipane for up to 2 days. Bring to room temperature before using.
Line a rimmed baking sheet with parchment paper. Cut the puff pastry into 6 individual 4-inch rounds, or use the shape and size of the pastry as purchased (10x13 inches) for one large tart. Place the pastry on the baking sheet. Using the tip of a knife, lightly score a 1/3-inch border. Prick the pastry with a fork all over. Spread the frangipane over the pastry to just inside the borders. Chill for at least 30 minutes.
Position a rack in the middle of the oven and preheat to 400 degrees F.
Arrange the quince slices decoratively on top of the pastry, brush the fruit with some of the syrup, and bake the galette until the pastry is golden and puffed on the top and crisped on the underside, 20 to 30 minutes. Brush the fruit with some of the reduced poaching syrup and serve warm with ice cream, and drizzled with more syrup, if desired.