To prepare yogurt cheese, place a colander in a 2-quart glass measure or medium bowl. Line colander with 4 layers of cheesecloth, allowing cheesecloth to extend over outside edges. Spoon yogurt into colander. Cover loosely with plastic wrap, and refrigerate 12 hours. Spoon yogurt cheese into a bowl; discard liquid. Cover and refrigerate.
To prepare ice cream, combine 1/4 cup sugar and eggs in a medium bowl, stirring well with a whisk. Combine milk and half-and-half in a heavy saucepan over medium-high heat. Scrape seeds from vanilla bean, and add seeds and bean to milk mixture. Heat to 180° or until tiny bubbles form around edge of pan, stirring frequently (do not boil). Gradually add milk mixture to egg mixture, stirring constantly with a whisk. Return milk mixture to pan; cook over medium heat until thick (about 5 minutes), stirring constantly. Remove from heat. Place pan in a large ice-filled bowl until mixture cools to room temperature (about 25 minutes), stirring occasionally. Remove vanilla bean. Pour mixture into the freezer can of an ice-cream freezer; freeze according to manufacturer's instructions. Spoon ice cream into a freezer-safe container; cover and freeze 1 hour or until firm.
To prepare the sauce, place 1/3 cup sugar in a small, heavy saucepan over medium-high heat; cook until sugar dissolves, stirring as needed to dissolve sugar evenly (about 6 minutes). Continue cooking about 1 minute or until golden. Remove from heat; carefully stir in water (caramelized sugar will seize slightly and stick to spoon). Place pan over medium-high heat until caramelized sugar melts. Scrape seeds from vanilla bean; add seeds and bean to mixture. Cook 2 minutes or until reduced to 6 tablespoons, stirring occasionally. Remove from heat, and let stand 10 minutes. Discard vanilla bean.
Place 1/4 cup yogurt cheese into each of 6 bowls; top with 1 tablespoon sauce and 1/2 cup ice cream. Serve immediately.