Preheat the oven to 350°F and line a couple of baking sheets with parchment paper.
In a large mixing bowl, stir together the water and ground flaxseed. Let the mixture sit for at least five minutes, to gel. You can proceed to the next step of mixing the dry ingredients in the meantime.
In a second large mixing bowl, whisk together the flour, baking soda, cornstarch, and salt.
Once the flax mixture has sat for at least 5 minutes, add the vegan butter, peanut butter, brown sugar, sugar, vanilla, and apple cider vinegar. Beat the mixture at high speed using an electric mixer for about 1 minute, until it lightens in color a bit and becomes fluffy.
Begin beating the flour mixture into the butter mixture, adding about a third at a time and beating each addition in with the mixer at low speed.
Stir 1 ½ cups of the chopped peanut butter cups and all of the chocolate chips.
Drop the dough on the cookie sheets in rounded mounds of 2 heaping tablespoons each, spacing them about 2 inches apart.
Bake the cookies for 11 to 12 minutes, until they've spread out and darkened very slightly around the edges. They will look undercooked and not fully spread out when they're done, with a little bit of doming at their centers. This is fine and will settle as they cool.
Place the baking sheets on cooling racks and gently press the remaining pieces of peanut butter cups into the surface of the cookies.
Let the cookies cool for about 10 minutes on the baking sheets, before removing them and transferring them directly to the racks to finish cooling.