Beat together eggs; half-and-half, light cream, or whole milk; sugar; cocoa; and vanilla in a large bowl with a rotary beater or wire whisk. Stir in rice and chocolate. Pour custard mixture into a 1-1/2- or 2-quart casserole. Place dish in a 13x9x2-inch baking pan set on an oven rack. Carefully pour 1-inch of boiling water into the baking pan.
Bake, uncovered, for 60 to 65 minutes or until a knife inserted near center comes out clean.
Stir together brown sugar and cornstarch in a saucepan. Stir in water, chocolate syrup, and molasses. Cook and stir mixture over medium-low heat for 2 minutes more or until thickened and bubbly.
To serve, spoon warm pudding into bowls. Pour 1 to 2 tablespoons sauce over each serving. Makes 6 to 8 servings.