Preheat the oven to 350°F, and line a large baking sheet with parchment paper.
In a medium bowl, sift together the almond flour, tapioca starch, baking soda, baking powder, and salt. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment or a large mixing bowl with a hand mixer, cream the butter and sugars on medium-high speed until light and fluffy, about 3-4 minutes, scraping down the sides as needed.
Add the peanut butter and mix until combined, scraping down the sides.
Add the egg and vanilla, and mix until combined, again scraping down the sides.
With the mixer on low, slowly add the dry ingredients and mix until just combined. Use a cookie scoop or rolling with your hands, scoop 2 Tablespoon-sized balls of dough, 2-inches apart on the prepared baking sheet(s). Use the tines of a fork to gently press down on the cookie in two directions, creating a criss-cross pattern.
Bake for 10-12 minutes until the edges are set and slightly firm to the touch, but the center still looks very soft.
Remove from oven and, if desired, sprinkle with flaky salt. Let cool for 5 minutes on the baking sheet before using a thin spatula to carefully transfer them to a wire rack to cool completely.
Store in an airtight container for several days, though the cookie edges will start to soften over time.