Heat oven to 350 degrees. Mix together flour, sugar, cocoa, baking soda and salt in a mixer bowl. Add egg, oil and vanilla; slowly beat in milk until combined, about 2 minutes.
Drop batter by tablespoons onto parchment-lined baking sheets, leaving room between each for spreading. Bake 12 minutes. Cool completely on wire racks.
For filling, combine all ingredients in a bowl; beat until smooth. Spread about 2 tablespoons of the filling per cookie on the flat side of half of the cookies. Top with remaining cookies to make sandwiches.