Combine the peaches, 2 tablespoons brown sugar, lemon juice, cornstarch, and 1/2 teaspoon cinnamon in a saucepan. Cook over medium heat for 5 to 7 minutes until thick, then cool completely.
Mix the streusel flour, 1/4 cup brown sugar, 1/2 teaspoon cinnamon, and melted vegan butter until crumbs form. Refrigerate while preparing the dough.
Beat the softened vegan butter, granulated sugar, and 1/2 cup brown sugar until light and creamy. Beat in the egg and vanilla.
Stir the 2 cups flour, baking soda, and salt into the butter mixture just until a soft dough forms. Refrigerate the dough for 30 minutes.
Preheat the oven to 350 degrees F and line two baking sheets with parchment paper. Divide the dough into 12 equal portions and roll each portion into a ball.
Place the dough balls 3 inches apart and make a wide indentation in each center. Fill each indentation with about 2 teaspoons peach filling.
Sprinkle the chilled streusel over the cookies. Bake for 14 to 17 minutes, until the edges are golden and the centers are set.
Cool the cookies on the baking sheets for 10 minutes, then transfer them to a cooling rack.
