Preheat the oven to 375º.
Combine the flaxseed meal and water in a small bowl. Set aside for 5 minutes. The flaxseed meal will absorb the water and thicken. In vegan cooking, this is often called a flaxseed egg.
In a medium-sized mixing bowl, combine the butter, peanut butter, brown sugar, white sugar, vanilla and oat milk. Use a hand mixer to thoroughly combine the ingredients. Add the flaxseed eggs and stir to combine.
In a large mixing bowl, combine the flour, salt and baking soda. Stir to combine.
Combine the wet ingredients into the dry ingredients. Stir to combine.
Line a baking sheet with parchment paper or silicon baking mats. Spread ½ cup of sugar on a small plate.
Take a large spoonful of cookie dough and roll it in your hands to make a ping-pong sized ball. Roll the ball in the sugar and place it on the baking sheet, keeping the dough balls about 2 inches apart on the baking sheet. Continue this process until you have used all of the dough.
Bake the cookies for 12 minutes on the middle rack.
Remove the baking sheet and add the blocks of chocolate to the center of each cookie. Press the chocolate block into the cookie, causing the cookie to crack slightly and secure the chocolate in the depression.
Return the baking sheet to the oven for 3 minutes.
Allow the cookies to cool on a cooling rack and store in an airtight container with a piece of parchment paper in between each layer of cookies.