To make the ice cream: Combine raisins and rum in a small glass bowl. Heat in the microwave for 2 minutes and then allow to stand until room temperature, about 15 minutes.
Reserve 2 tablespoons of rum raisin liquid and drain the raisins and set them aside. In the bowl of a stand-up mixer with the whisk attachment, combine the sweetened condensed milk, heavy cream and pinch of salt. Beat for about 2 to 3 minutes, until soft peaks form. Do not walk away because it’s so easy to over-beat this mixture. Fold in the rum and raisins. Transfer to a 8x4-inch loaf pan or any other 3-quart freezer-safe container. Cover with plastic wrap and place in the freezer until firm, about 8 hours.
To make the buns: In a medium bowl or 4-cup measuring cup, combine the milk, sugar and yeast. Allow to stand for 5 minutes, until foamy on top. This will let you know the yeast has been activated. Whisk in the egg and egg white.
Meanwhile, in the bowl of a stand-up mixer with the dough hook, add the flour, cinnamon and salt. Mix for a few seconds until flour mixture is combined. Pour in the egg and milk mixture and mix for about 30 seconds, until most of the liquid is absorbed into the flour mixture. Add the butter and turn the mixer to medium speed and allow the dough to knead for about 5 minutes. The dough will be soft and slightly sticky. Cover the bowl with a clean kitchen towel and place in a warm area of your home to rise for 1 hour.
Spray a 9 x 13-inch baking dish with a bit of cooking spray. Divide the dough into 12 pieces (that weight about 2.3 ounces each) and shape them into balls. Place them side by side in the baking dish and cover with a clean kitchen towel. Allow them to rise for about 30 to 45 minutes.
Meanwhile, preheat your oven to 375 degrees F. Brush the tops of the balls with the egg wash mixture and transfer to the oven to bake for 20 minutes. Allow to cool in the pan for 5 minutes and then transfer to a wire rack to cool.
To make the frosting: whisk the sifted powdered sugar with the milk and vanilla extract. Transfer to a piping bag or 1 gallon ziplock and snip the end with a pair of scissors.
To assemble the ice cream sandwiches: Slice the buns in half. Pipe the frosting onto the tops and place a scoop of ice cream onto the center of each bottom half. Place the top bun on top and eat away!