Warm the milk in a 2-quart heavy-duty saucepan over medium heat until small bubbles form around the edges.
Whip together the sugar, egg yolk and salt in the bowl of an electric stand mixer with the wire whip attachment or in a large mixing bowl using a hand-held mixer on medium speed until the mixture is very thick and pale yellow and holds a slowly dissolving ribbon when the beater is lifted.
Ladle half of the hot milk into a liquid measuring cup, then slowly pour it into the egg yolk mixture, whisking constantly to blend well. Pour this egg mixture into the remaining milk in the saucepan. Stir in the candied ginger puree. Whisk the mixture constantly until it is thick enough to coat a spoon or registers 185 degrees F. on a candy thermometer. Immediately remove the thermometer and pour the ice cream custard into a 2-quart bowl. Add the cream and vanilla and stir to blend thoroughly.
Cover the bowl tightly with plastic wrap to prevent a skin from forming on top and cool to room temperature. Refrigerate the ice cream custard for several hours, or until it is cold.
Process the ice cream custard in an ice cream maker, following the manufacturer’s instructions. When the ice cream is half frozen, stir in the finely chopped crystallized ginger. Freeze the ice cream in a tightly covered container for at least 30 minutes.
Using a 1-1/2 inch round ice cream scoop, place small scoops of ice cream in shot glasses, mini martini glasses or small bowls. Top each serving with a sliver or two of crystallized ginger.
The ice cream custard can be made up to 3 days in advance and kept tightly covered in a bowl in the refrigerator before freezing in the ice cream maker.
Keep ice cream in a tightly covered container in freezer for up to 2 weeks.