In a bowl, beat the butter, sugar, molasses and egg with an electric mixer.
In another bowl, combine the flour, baking powder and ginger.
With a wooden spoon, add the dry ingredients to the previous mixture alternately with the milk. Cover with plastic wrap. Refrigerate for 1 hour.
Preheat the oven to 180 °C (350 °F). Line a baking sheet with parchment paper.
Divide the dough into thirds. Keep two of the three parts in the refrigerator. On a floured work surface, roll out the dough thinly, like a pastry crust. Cut into 6 x 9-cm (2 1/2 x 3 ½-inch) rectangles, keeping the scraps to make cookies. Place on the baking sheet. Bake in the middle of the oven for about 8 minutes.
Let cool. Repeat with remaining dough.
Remove the packaging from around the ice cream. Cut a 1.25-cm (1/2-inch) thick slice, cut in half to create two rectangles the size of the cookie. Place a portion of ice cream between two cookies. Gently squeeze and wrap each sandwich in plastic wrap. Freeze immediately. Repeat with the remaining ingredients.
For children, you can decorate the edges of sandwiches with candy sprinkles or unsalted peanuts, toasted and chopped. Do this just before serving, as the candy tends to discolor.
Recipes with the "Healthy Pick" stamp have been evaluated by a registered dietitian member of the Ordre professionnel des diététistes du Québec.