Make the cakes: Preheat the oven to 350 degrees . Line 2 baking sheets with parchment. In a large bowl, whisk together 1 1/2 cups granulated sugar, 2 eggs, the cocoa, baking soda, 1 tsp. salt and 1 tsp. vanilla until slightly thickened.
In another bowl, briefly whisk together the oil, food coloring and vinegar. Whisk the oil mixture into the sugar mixture. Whisk in 1 cup flour, then 1/2 cup buttermilk; repeat. Finally, whisk in the remaining 1 cup flour.
Dollop the batter 2 inches apart on the prepared baking sheets. Bake the cakes, switching the pans halfway through baking, until the tops are set and just springy, 11 to 12 minutes. Let cool.
Make the ice cream: Using an electric mixer fitted with the whisk attachment, beat the egg whites and remaining 1/4 tsp. salt on low until foamy. Increase the speed to medium and gradually beat in the remaining 1/3 cup granulated sugar until the meringue has just formed stiff peaks. Transfer the meringue to a bowl and refrigerate.
Add the cream cheese, confectioners' sugar and remaining 1/2 tsp. vanilla to the mixer bowl. Switching to the paddle attachment, mix on low until combined. Increase the speed to medium-high and beat until light and fluffy, 2 to 3 minutes. Scrape down the bowl as needed.
Beat about one-third of the chilled meringue into the cream cheese mixture on low speed. Add half of the remaining meringue and gently fold in with a spatula. Repeat with the last bit of meringue. Scrape the mixture into a container, cover and freeze overnight.
Sandwich the ice cream between the red velvet cakes. Serve immediately.