Marie Louise's Rice Pudding

I wonder if I can make this with glutinous black rice.

Serves 4 to 6

3/4 cup long-grain rice
1 bay leaf
6 cups whole milk, approximately
1 cup sugar
1/2 vanilla bean, split lengthwise
Large pinch of salt

1. Place rice in a small saucepan with bay leaf and 2 cups water, bring to a boil over high heat, then drain immediately. Transfer rice and bay leaf to a heavy 3-quart saucepan.

2. Add 4 cups milk, 1/2 cup sugar, vanilla bean and salt. Place over very low heat and cook, stirring occasionally, until milk has been absorbed by rice, about 1 hour. Add 1 or 2 more cups milk, and continue cooking over low heat 15 to 20 minutes longer. Rice should be tender and mixture should be very creamy. Remove from heat, and allow to cool to room temperature.

3. Combine remaining sugar with 2 tablespoons water in a 1-quart saucepan. Place over medium-high heat, and stir gently until sugar dissolves. Continue cooking until mixture turns a medium amber color. Remove pan from stove and place on a heat-proof surface to cool. Caramel will darken somewhat and harden as it cools.

4. No more than 30 minutes before serving, add a little milk to rice if it has become too thick, then remove bay leaf and vanilla and transfer pudding to a shallow serving bowl. Add 2 tablespoons water to caramel, return it to stove and place over medium heat. Cook about 5 minutes until caramel has softened. Stir to incorporate water. Remove from heat. To serve, drizzle caramel sauce over pudding.