In a bowl, whisk together flour, baking soda, and salt. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, or a large bowl with an electric hand mixer, beat butter until smooth. Cream the butter with 1/2 cup granulated sugar and the light brown sugar until light and fluffy. Beat in peanut butter.
Scrape the bottom and sides of the mixing bowl. Beat in milk and vanilla, followed by the egg, until combined.
Add the flour mixture and mix until a stiff dough forms. Cover the mixing bowl and refrigerate for 30-60 minutes.
Preheat oven to 375 degrees F. Place 1/2 cup granulated sugar in a shallow bowl.
Scoop the dough into 1-1/4 inch portions (40-42 portions), roll them into balls, and then roll them in the granulated sugar. Place on ungreased baking sheets, 2 inches apart.
Bake the cookies 8-10 minutes, until set, lightly brown, and just starting to crack. Don't overbake.
As soon as you remove the cookies from the oven, gently press a Hershey's kiss into each center. (The cookies will crack - this is normal and part of their classic appearance.) After about 5 minutes, transfer the cookies to a wire rack to cool completely, at least an hour. (The chocolate should be firm before the cookies are stored.)
Store at room temperature in an airtight container for 3-5 days, or freeze for 2-3 months.
*Inspired by Mrs. Freda F. Smith's recipe, a finalist in the 1957 Pillsbury Bake Off® contest. My family's variation swaps butter for the shortening in the original recipe, and adds additional peanut butter.