Press cold pastry over bottom and up sides of a 9-in. tart pan with a removable rim and trim edges even with top of pan. Chill 30 minutes. Meanwhile, preheat oven to 375°.
Bake pastry on bottom rack of oven until golden brown, 25 to 30 minutes. Let cool completely.
Stir frozen yogurt until smooth, then stir in preserves. Spoon into tart pan, set on a plate, and freeze at least 5 hours.
Simmer balsamic vinegar, sugar, and 1 tbsp. water in a small saucepan over medium heat (do not let boil) until it coats a spoon, about 10 minutes. Let cool.
Top pie with strawberries. Thin syrup with water if necessary and drizzle onto each wedge of pie.
More pie tips:
Let the pie soften for 5 minutes at room temp to make slicing easier.
If you're having trouble freeing your pie from its pan, set it over a bowl of hot water for a couple of minutes and then slide a thin knife between the pan edge and the crust. It should pop right out.
Make ahead: Once the pie is fully frozen through step 3, it keeps for up to 4 days, double-wrapped in plastic wrap. Make the syrup and top the pie just before serving.