Stir the milk and ground flaxseed together in a small bowl. Set the mixture aside for at least five minutes, to give the seeds time to gel.
While the flax mixture sits, beat the butter, sugar, brown sugar, and vanilla together in a large mixing bowl, using an electric mixer at high-speed and scraping down the sides of the bowl as needed. Beat the ingredients just until combined, about thirty seconds.
Add the flax egg to the bowl, then beat it into the butter mixture, just until combined.
Add the peanut butter to the bowl, then beat it in for about one minute, until the mixture is creamy. Continue scraping down the sides of the bowl as needed.
Turn the mixer down to low speed and begin adding the flour in increments, beating in about a third at a time. Sprinkle the baking soda and salt over the last addition of flour before beating it in. Stop beating the mixture once the ingredients are fully combined and a fluffy dough is formed.
Cover the bowl and chill the dough for one hour, until it has firmed slightly, but isn't yet rock hard.
Preheat the oven to 350°F and line a couple of baking sheets with parchment paper.
Make each cookie using one heaping tablespoon of dough and rolling it into a ball. Arrange the dough balls on the baking sheets, leaving at least an inch and a half between cookies. Flatten each dough ball with a fork by first pressing the back of the fork into one of the dough balls, then rotating the fork 90 degrees and pressing again, to create a grid.
Place the baking sheets into the oven and bake the cookies for 10 to 12 minutes, until just barely set.
Transfer the baking sheets to cooling racks. After about five minutes you can remove the cookies from the baking sheets and transfer them directly to the cooling racks. The cookies will continue to set as they cool.