Preheat the oven to 400°F. Add the plums to a medium bowl and toss with ⅓ cup sugar and cornstarch. Set aside.
Prepare the caramel: Cook ¼ cup sugar, butter, and salt in a medium pot over medium heat, swirling the pan occasionally, until it is an even deep amber, about 4 to 6 minutes. Remove the pan from heat and carefully add the cream and the anise seed (take care: the mixture may sputter). Stir to combine, then transfer the caramel to and a bowl and cool to room temperature.
Roll the pie crust out into a 13-inch circle and transfer to a parchment-lined baking sheet. (Refrigerate the dough for a few minutes if begins to get soft.) Spread the caramel in the center of the circle, leaving a 3½-inch border on all sides. Top the caramel with the plums and any juices. Fold the edges of the dough up and partially over the fruit, overlapping to create a 3-inch border. Freeze galette for 15 minutes.
Brush the top of the dough with the egg. Bake the galette until the fruit is tender, the juices are bubbling, and the crust is deep golden brown, 40 to 50 minutes. (Tent with foil if the crust browns too quickly.) Serve warm or at room temperature.