In a medium bowl whisk together flour, baking soda and salt. Set aside.
In a mixer bowl fitted with the paddle attachment, beat butter, peanut butter, granulated sugar and brown sugar on medium speed for 3-4 minute until fluffy and creamy. Add the egg, milk, and vanilla extract, and beat until combined. Add flour mixture and mix just until combined. Do not overmix.
Place 1/4 cup granulated sugar in a small bowl for the topping. Set aside. Roll dough into balls, the size of 1.5 tablespoons each (I use this ice cream scoop for even cookies), and 1.5 inches/4 cm in diameter. Roll balls in sugar, and place onto the cookie sheets, 3 inches/8 cm apart.
Bake cookies for 10 minutes until puffy and slightly cracked. Allow cookies to cool for 5-10 minutes, then gently press peanut butter cup into the center of each cookie. Cookies will crack, that’s ok. If they break too much, squeeze the broken-off bit back onto the cookie. The peanut butter cups will slightly melt and that’s ok (you can place the cookies in a plate and in the freezer for several minutes, but that’s optional). Gently transfer cookies to a wire rack to cool completely. Serve cookies warm or at room temperature.
Store cookies at room temperature in an airtight container for up to 3 days. If you plan to keep them for longer, place in the fridge for up to 1 week, or freeze for up to 2 months.