Heat 4 cups of the milk in a large saucepan over medium heat until steaming.
Whisk the sugar, cornstarch and 1/2 teaspoon salt in a large bowl. Whisk in the egg yolks and the remaining 1/2 cup milk.
Whisk half of the hot milk into the egg mixture until smooth, then gradually whisk the egg-milk mixture into the saucepan. Cook over medium heat, whisking constantly, until the mixture boils. Continue to cook, whisking constantly, until it has thickened to a puddinglike consistency, 3 to 4 minutes. Remove from the heat and stir in the vanilla.
Cool slightly, stirring a few times to prevent a skin from forming. Place a piece of plastic wrap directly on top of the pudding and refrigerate until completely cooled and thickened, about 4 hours.
Before serving, whisk the pudding vigorously until smooth and creamy. Gently fold in the berries. Spoon the pudding into six to eight individual cups and top with the crushed gingersnaps.
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