Preheat the oven to 350°F. Lightly grease (or line with parchment) two baking sheets.
Using mixer, beat together butter, sugars, egg, vanilla, and peanut butter until smooth.
Add the flour, baking soda, and salt to the peanut butter mixture, beating gently until everything is well combined, no flour streaks.
Drop the cookie dough by tablespoonful or by using a 1 1/2 Tablespoon cookie scoop, onto the prepared baking sheets leaving 2" in between each cookie.
Gently use a fork to flatten each cookie to about 1/2" thick, making a cross-hatch design.
Bake the cookies for 12 to 16 minutes, until they're barely beginning to brown around the edges to get best results for soft and chewy peanut butter cookies. Bake longer for crunchy. Remove them from the oven and let set to cool on baking sheet for 5 minutes to set up so they don't fall apart.
Transfer to a wire rack until cooled.
Place sprinkles in a shallow bowl and set aside.
Add one heaping scoop of softened ice cream to flat side of one cookie.
Add second cookie to the top, flat side down, then push down slightly to seal cookies and ice cream together to make a sandwich.
If ice cream is too soft and squishes out, use a small spatula to scrape around the ice cream and smooth the edges.
Dip edges of the ice cream sandwich in the sprinkles. Individually wrap sandwich in plastic wrap to freeze for a few hours.
Yield: about 2 dozen cookies, makes 12 sandwiches