Freeze the banana pieces overnight in resealable plastic bags.
Put the bananas, mascarpone, sugar, vanilla seeds, and lemon juice in a food processor. Pulse for a few seconds, and then whip until smooth, about 20 to 30 seconds. Lay 3-inch ring molds on parchment-lined baking sheets, and fill evenly with the mixture. Freeze until firm, 1 hour or longer, if needed.
Pop out of the molds, and place sandwiched between Double Chocolate Cookies.
Position an oven rack in the center of the oven and preheat to 350 degrees F. Line two baking sheets with parchment paper. Sift together the flour, cocoa powder, baking powder, baking soda, and salt in a large bowl.
In the bowl of an electric mixer or food processor, cream the butter and sugars until light and fluffy. Add the eggs and vanilla, and beat until combined. Add this mixture to the bowl of dry ingredients, and stir to combine.
Fold in the chips and nuts, until just combined. Drop by ice cream scoopfuls onto a parchment-lined baking sheet. Bake for 12 to 14 minutes, or until just cooked, but still soft. Set cookies onto a wire rack to cool.
Using the same mold rings as the ice cream, cut the cookies into 'sandwiches'. Assemble with the ice cream and serve.