Combine all-purpose flour flour, cane sugar, and sea salt in a large bowl.
Using a pastry cutter, work the vegan butter into the flour until it resembles coarse meal.
Mix vodka, ice water, and apple cider vinegar in a small bowl. Drizzle in the liquid mixture a tablespoon at a time into the flour mixture, tossing and mixing with a spatula or wooden spoon between each addition. Stop adding water when the dough comes together, and almost no more dry flour remains in the bottom of the bowl.
Gather the dough and wrap tightly with plastic wrap. Flatten into a 4 to 6-inch disk. Refrigerate for at least 90 minutes.
Let the dough warm up slightly before rolling it out, approximately 10 minutes.
Make the filling
Combine sliced quinces in a large mixing bowl with lemon juice. Place ice cubes in a layer on a large rimmed baking sheet and set aside.
Combine sugar, water, and agave syrup in a cast iron skillet or tarte tatin pan on medium heat. Stir until sugar dissolves.
Boil gently until sugar mixture is a golden color, about 10 minutes. Watch carefully. Remove skillet from heat; stir in vegan butter and apple pie spice or cinnamon, if using..
Place the skillet on top of ice in large rimmed baking sheet for 30 minutes until caramel hardens. Remove skillet from ice. Position a rack in center of oven; preheat to 400°F.
Layer quince slices in circles on top of the cold caramel in the skillet. Fill center with any small or broken quince pieces.
Flour dough lightly on both sides. Roll out dough between 2 layers of parchment paper to a 10 to 12–inch round, approximately ⅛-inch thick.
Place parchment paper dough side down on top of quinces and slowly peel away the parchment paper. Tuck in the dough on top and around sides of quinces.
Place skillet on the rimmed baking sheet. Bake until crust is deep golden brown, about 1 hour. Remove from oven; let cool 30 minutes before flipping.
Place a platter or serving plate on top of the skillet. Wearing oven mitts, hold the plate and skillet firmly together and turn over. Let the tart slide out onto the serving plate. Serve warm or at room temperature,
Notes
The dough can be made the day before; store it overnight in the fridge. You can use a food processor to make the dough instead of doing it by hand.Store the baked tarte tatin at room temperature for up to 2 days covered with plastic wrap or in an airtight container.Recipe adapted from Epicurious.