Preheat oven to 350 degrees F. In a large bowl, beat the 1/2 cup butter with an electric mixer on medium to high speed for 30 seconds. Add the 1/2 cup brown sugar, the granulated sugar, baking soda, and salt. Beat until combined, scraping side of bowl occasionally. Add the 3/4 cup peanut butter, the 1/4 cup peanuts, the egg, and 1/2 teaspoon vanilla; beat until combined. Beat in as much of the flour as you can with the mixer. Using a wooden spoon, stir in any remaining flour.

Shape dough into 3/4-inch balls. Place balls 2 inches apart on an ungreased cookie sheet. Flatten by making crisscross marks with the tines of a sugared fork. Bake in preheated oven for 7 to 9 minutes or until bottoms are light brown. Transfer cookies to a wire rack; let cool.

For filling: In a medium bowl, beat the 1/4 cup butter with an electric mixer on medium to high speed for 30 seconds. Add the 1-1/2 cups peanut butter, the powdered sugar, milk, the 2 tablespoons brown sugar, and 1/2 teaspoon vanilla. Beat until smooth and creamy.

Spread 1 teaspoon of the filling on bottom of each of half of the cookies. Top with remaining cookies, flat sides down, pressing lightly together. Place sandwich cookies on wire racks set over waxed paper.

In a small heavy saucepan, cook and stir chocolate over low heat until melted; cool slightly. Spoon melted chocolate over assembled sandwich cookies, allowing excess to drip down sides of cookies. If desired, sprinkle with the 1-1/2 cups peanuts. Let stand about 1 hour or until chocolate is set. Makes about 72 sandwich cookies.

To Store: Layer unfilled cookies between waxed paper in an airtight container; cover. Store at room temperature for up to 3 days or freeze for up to 3 months. To serve, thaw cookies if frozen. Prepare filling as directed in step 3. Assemble and decorate as directed in steps 4 and 5.