Rinse and stem strawberries; cut each in half lengthwise. Rinse rhubarb well; trim off and discard ends. Slice stalks crosswise into 3/4-inch pieces.
In a 3- to 4-quart pan, mix 3/4 cup sugar and cornstarch. Add berries and crush with a potato masher. Cut vanilla bean in half lengthwise and scrape seeds into pan. Add vanilla pod, orange peel, cinnamon, and rhubarb. Cover and cook over medium heat, stirring occasionally, until rhubarb is soft when pierced, 15 to 20 minutes. Taste, and add more sugar if desired.
Remove vanilla pod. Spoon compote into bowls. Top with scoops (1/2 cup each) of ice cream and Lattice Cookies.