Preheat the oven to 350ºF and line a baking sheet with a silicone baking mat or parchment paper.
In a large food processor fitted with an "S" blade, process the dates until crumbly. (It's okay if a ball forms, too.)
Add in the peanut butter, ground flax, water, vanilla, baking soda, and vinegar. Process until a sticky, uniform batter is created. Be careful not to over-process the mixture, as the peanut butter will release excess oil when over-mixed, and that will make the batter feel very oily to work with. (Don't worry if this happens, though-- the cookies still turn out totally fine!)
Use a heaping tablespoon to scoop the dough into your hands, then roll the dough into a ball between the palms of your hands. Repeat with the remaining batter, making about 20 balls. Place each ball onto the baking sheet about an inch apart, then use your hands to gently flatten each ball into a cookie shape. Sprinkle a bit of coarse sea salt over each cookie, if desired, then bake until the edges of the cookies start to feel dry, about 13 to 15 minutes. (The cookies will still feel fragile at this point, but that's okay.) The longer you bake the cookies the more firm they will be, so at 13 minutes of baking you'll have a soft, almost under-baked middle (which is how I love them!) and at 15 minutes they will be a little more firm and brown around the edges. I imagine you could bake them up to 20 minutes without burning them, if you want a cookie that is even more firm, but my family likes our cookies to be soft and tender in the middles.
Allow the cookies to cool completely before removing them from the baking sheet and serving. Leftover cookies can be stored in an airtight container in the fridge or freezer. (The cookies will soften if you leave them out on the counter overnight, so chilling them is best for texture and shelf life.)