Preheat oven to 375 degrees.
In a small bowl, whisk together ground flaxseed and water.
In a large bowl, using a stand or handheld mixer on medium speed, cream together flaxseed and water, peanut butter, vegan butter, vanilla extract, granulated sugar, and ½ cup light brown sugar.
In a medium size bowl, stir together flour, baking soda, and salt. Add to the peanut butter mixture and continue mixing until fully incorporated. Dough will be thick and crumbly.
Working in batches, form dough into 1” balls and roll each piece in the reserved 1/3 cup light brown sugar.
Place 3” apart on a non-stick baking sheet. Using the tines of a fork, lightly press a criss-cross pattern into each ball of dough.
Bake in preheated oven for 9 minutes. Remove from oven and allow to cool on baking sheet for 2 minutes, then place on a wire rack to finish cooling completely. Store in an airtight container.