Make the cookies: Preheat the oven to 375 degrees F. Line 2 large baking sheets with parchment paper. Combine the semisweet and milk chocolate in a microwave-safe bowl and microwave at 50 percent power until melted, about 2 minutes. Whisk until smooth.
Whisk the melted butter, sour cream, eggs, vinegar, vanilla and food coloring in a bowl until combined. In another bowl, whisk the flour, sugar, cocoa powder, baking powder, baking soda and salt.
Add the flour mixture to the butter mixture in four equal batches, whisking each batch completely before adding the next. Stir in the melted chocolate.
Scoop heaping tablespoonfuls of batter onto the prepared baking sheets and smooth the tops with a damp finger. Bake until the cookies spring back when lightly pressed, 8 to 10 minutes. Let cool 10 minutes on the baking sheets, then transfer to racks to cool completely.
Meanwhile, make the filling: Beat the cream cheese and butter with a mixer until smooth. Beat in the confectioners' sugar and vanilla seeds. Sandwich a heaping tablespoonful of filling between 2 cookies; repeat with the remaining cookies and filling. Refrigerate 30 minutes before serving.