Position rack in center of oven and preheat to 350°F. Line 2 baking sheets with parchment paper. Stir butter, sugar, and corn syrup in heavy medium saucepan over low heat until melted and smooth. Bring to boil over medium-high heat, stirring constantly.
Remove from heat. Stir in flour.
Add nuts and vanilla; stir to combine.
Drop dough by teaspoonfuls onto prepared baking sheets, spacing 2 inches apart.
Bake cookies, 1 sheet at a time, until bubbling and lightly browned, about 11 minutes. Cool on sheets 10 minutes.
Transfer to rack; cool completely.
Whisk all ingredients in medium bowl until smooth.
Spread 1 teaspoon filling onto bottom of 1 cookie. Top with second cookie, bottom side down, pressing lightly to adhere. Repeat with remaining cookies. (Can be made 2 days ahead. Store between sheets of waxed paper in airtight container at room temperature.)