Roll out the larger piece of dough (keep second piece refrigerated) into a 13-inch round. Fit into a 9” pie plate. (If you will later use a decorative pastry stamp for the top crust like I did, fold and crimp the edges). Refrigerate (or pop in the freezer if your fridge is too full like mine).
Preheat oven to 400 degrees with the rack in the middle of the oven. Place a lipped baking sheet lined with foil on the rack to catch any juices that might bubble over.
Combine strawberries with granulated and brown sugars, followed by balsamic vinegar, lemon juice and vanilla extract. Stir in the salt and corn starch, and then add the black pepper. Let the fruit filling sit for at least 20 minutes.
Roll out the dough for the top crust into an 11-inch round. (If using a pastry stamp, now is the time to make cutouts.)
Toss the berries gently and spoon into refrigerated shell. Cover with the top pastry round (or decorate top with stamped shapes). Press or fold edges to seal and crimp decoratively. Brush top of pie with milk or cream, cut steam vents, and sprinkle crust with sugar.
Put pie in oven and bake for 20 minutes. Reduce oven temperature to 375 degrees, and bake until crust is deep golden and filling is bubbly in the center, 20-35 minutes more.
Cool completely on a rack. Enjoy with family and friends (with a nice cappuccino).