Pre-heat the oven to 375F.
Pour 1 ¼ cup of strained fruit poaching liquid into a nonstick skillet. Cook over medium heat until the liquid is thick and syrupy (the consistency of honey), and then remove from heat. You should have ¼ cup left.
Pat the quince quarters dry, and lay the poached quince snugly against each other, rounded side down, in the pan. Pack them in tightly as they'll settle down once baked.
Remove the dough from the fridge. On a lightly-floured surface, roll the dough to ⅛" thick and into a circle a few inches bigger than the pan you're using. Only roll as much dough as you'll need for this recipe. Refrigerate leftover dough for another dish.
Drape the dough over the quince, tucking in the edges, and bake on a lower oven rack for 45 minutes. The tart is done when the dough is deep golden brown.
Remove the Quince Tarte Tatin from the oven and let it rest for a few minutes. Place a rimmed serving platter or baking sheet over the tart. Flip the tart over. You may wish to wear long oven mitts and be sure to take appropriate precautions as hot liquid will inevitably escape. Save this hot liquid to glaze the tart!
Serve with yogurt or whipped cream.