Preheat the oven to 350ºF.  Line a baking sheet with parchment paper and set aside.  In a medium bowl, whisk together the flour, cocoa powder, soda and salt.  Set aside.

In a large bowl, or the bowl of a stand mixer, beat the butter and sugar together until light and fluffy, about 2 to 3 minutes.  Add the egg, food coloring and vanilla and beat to incorporate.  Alternate the flour and buttermilk into the butter mixture and beat until smooth.

Using a hinged ice cream scoop (or 1/4 cup measure) drop cookie batter onto the prepared baking sheet about 2 inches apart as the batter will spread.  Bake 12-15 minutes, or until the centers spring back when touched lightly.  Cool for 5 minutes on the baking sheet and then transfer to a wire rack and cool completely.

Blend cream cheese, milk, lemon juice, vanilla, sugar and salt in a blender or food processor until smooth.  Transfer to a bowl and stir in the cream.

Using an ice cream maker, freeze according to manufacturers instructions.  While the ice cream is churning, mix the melted white chocolate and food coloring together with the cream.  When the ice cream is almost done, pour in the chocolate mixture.  Transfer to an airtight container and put in the freezer to harden, about 2 hours.

Remove ice cream from freezer and allow to sit for 5 minutes before assembling.

Using a hinged ice cream scoop, add about 1/4 cup on a cookie and place another on top and press gently to spread the ice cream out to the sides.  Place on a cookie sheet and continue with the remaining 9 cookies.  Put the cookie sheet into the freezer to set the sandwiches for about 1 hour.  Remove and wrap each individually with plastic wrap.  Put back in the freezer until you're ready to eat them.