Combine peanut butter and pecans in a food processor and process until smooth. Whisk flour, baking soda, and salt in a small bowl.
Beat both sugars and butter with an electric mixer in a large bowl until fluffy. Add peanut butter mixture and vanilla and beat to blend well. Beat in egg. Add dry ingredients and mix until combined. Add both oats. Continue to blend at high speed for 1 minute.
Line 2 baking sheets with foil. Scoop balls of dough onto cookie sheets, spacing 1″ apart. Bake until light golden, 10 minutes.
Let cookies cool on sheets for a couple minutes, then transfer to wire racks to cool completely.
For frosting, mix all ingredients with an electric mixer. Spread one cookie at a time with the frosting and top with another cookie.