To make the crust: Combine the flours, 3 tablespoons of the sugar, and salt, if using, in a large bowl; whisk to blend well. Cut the butter or vegan butter into ½-inch pieces. Add the butter or vegan butter to the flour mixture and work in with your fingertips until a coarse meal forms (do not overwork the dough; some large bits can remain).
Using a fork, stir in just enough of the ice water to form a dough. Knead the dough briefly until it comes together. Pat the dough into a round disk, cover with plastic wrap, and refrigerate it until you’re ready to roll it out.
To make the filling: Preheat the oven to 425° F. Place the strawberries in a medium bowl. Sprinkle the strawberries with the sugar, cornstarch, and lime juice; toss to coat.
Place a large sheet of parchment paper on a work surface; sprinkle it with flour. Place the dough disk in the center of the parchment paper and, using a rolling pin, roll it out to a 13-inch round.
Lift the parchment paper up and transfer it, with the crust, to a baking sheet.
Spoon the strawberries into the center of the dough, drizzling with a bit of the juices from the bottom of the bowl. Fold about 1½ inches of the edges of the tart dough in toward the center, partly covering the strawberries.
Sprinkle the top crust around the edges with the remaining 1 teaspoon of sugar. Bake the tart until the crust is golden brown and the filling is bubbling, about 25 minutes.
Let stand for at least 15 minutes (the center will be very juicy but it will set some as it cools). Serve warm or at room temperature.