For the vanilla ice cream: Whisk the eggs in a large mixing bowl. Whisk in the cream and milk, a little at a time, until incorporated; then add the sugar, a little at a time, whisking until completely dissolved.
Pour the mixture into a glass bowl set over a pot of simmering water and cook, stirring occasionally, until thickened enough to coat the back of a spoon, about 40 minutes. Remove from the heat and stir in the vanilla. Let the ice cream base cool completely before adding it to an electric ice cream maker. Freeze according to manufacturer's instructions.
For the root beer float: In a frosty mug, add 2 large scoops of the vanilla ice cream, then top off with root beer. Repeat with the remaining ice cream and root beer. Serve with large straws and spoons.