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Adjust oven rack to middle position. Preheat oven to 350 degrees.
Sift flour into bowl of a food processor (scoop and sweep method if not using a scale).
Add peanuts and pulse until fine (approximately 1 minute).
Combine peanut butter, butter, sugar, salt, baking soda, baking powder and vanilla in bowl of a stand mixer fitted with paddle attachment.
Mix on low speed to moisten then increase to medium and beat until soft and light, about 3 minutes.
With mixer running add the egg in 2 additions, mixing until each is well incorporated.
Reduce speed to low and add peanut flour, followed by milk, mixing to form a very soft dough.
Divide into 34 2 Tbsp (1.125 ounce) portions.
Arrange on parchment lined baking sheet, 2 inches apart.
Bake until the edges are firm and just beginning to brown, but cookies are still puffy and steamy in the middle, about 16 minutes.
Cool on baking sheet until cookies are set, about 10 minutes.
Store in airtight container for up to 1 week at room temperature.