In a small bowl, beat the egg whites with an electric mixer until foamy. Gradually add 2 tablespoons sugar, beating until soft peaks form, and reserve.
In a saucepan, whisk together 2 tablespoons of sugar, the tapioca, milk, vanilla, and egg yolk. Let stand 5 minutes.
Cook the mixture over moderate heat, stirring constantly, until it comes to a full boil. Remove the pan from the heat and immediately stir in the reserved egg white. Return the mixture to low heat and cook, stirring constantly, for 2 minutes. Cool the tapioca, its surface covered in plastic wrap, for 20 minutes.
In a nonstick skillet, melt the butter. Add the apricots and the remaining 1/2 teaspoon sugar and cook over moderate heat, stirring occasionally, for 2 to 3 minutes. Add the tablespoon of milk and stir to coat. Remove from the heat and stir in the raisins, cinnamon, and ginger. Divide the tapioca among 4 bowls and top with the apricot mixture.