Cook rhubarb with brown sugar in a medium frying pan over medium-low heat until it looks like jam, 20 minutes. Put in a bowl and chill. Stir in ice cream and freeze until firm, 3 hours.
Preheat oven to 350°. Line a baking sheet with parchment paper. Mix 1 tbsp. sugar and the cinnamon in a bowl. Beat butter, 5 tbsp. sugar, the vanilla, and salt in a bowl until light and fluffy. Add flour and mix on low until large clumps form.
Gather dough into a ball and roll into a 1/4-in.-thick round. Transfer to baking sheet, sprinkle with cinnamon sugar, and freeze until firm, 10 minutes. Bake until edges are golden brown, 15 to 20 minutes. Cool, then break into large shards.
Put berries, cornstarch, lemon juice, and remaining sugar in a small saucepan. Bring to a simmer over medium-low heat and cook, stirring, until thick and bubbly, 5 minutes.
Divide berry compote among 4 bowls. Scoop ice cream into each bowl and add crust shards.