Preheat oven to 350F. Lightly grease (or line with parchment) 2 baking sheets.
To prepare cookies, combine sugars, peanut butter, shortening, salt, baking soda and vanilla, beating until smooth. Add egg, beating to combine. Stir in oats and flour.
Drop dough by heaping teaspoonfuls onto prepared baking sheets about 1 1⁄2 inches apart. Using the bottom of a drinking glass dipped in sugar, flatten each cookie to about 1⁄4-inch thickness, about 1 1⁄2-inch diameter. Sprinkle with a tiny amount of extra-fine salt, if using.
Bake 12 to 14 minutes, until medium golden brown. Remove from oven and transfer to a wire rack to cool. When cool, wrap airtight; they’ll soften if left unwrapped.
To prepare peanut butter filling, combine all ingredients, stirring until smooth. 6. Spread filling on half the cookies. Top with remaining cookies. Makes about 22 sandwich cookies.