In large bowl, whisk together flour, cardamom and salt. Using hands, crumble butter into flour until it has a sandy consistency.
Add icing sugar and vanilla sugar; stir in egg and quickly knead dough together. Wrap dough in plastic wrap, press into a disk and refrigerate for 1 hour.
Preheat oven to 375°F.
On lightly floured work surface, roll out dough to ⅛-inch thickness; using small, round cookie cutter or glass, cut out 32 shapes and place on two parchment-paper-lined baking sheets.
Bake 1 sheet at a time until slightly golden, about 10 minutes. Let cool on rack.
Using hand mixer, beat egg white in bowl until silky; beat in icing sugar, little by little, until mixture has a toothpaste-like consistency. Beat in vinegar at the end.
Spread icing on flat side of half the cookies. Spoon 1 tsp rhubarb compote on flat side of remaining cookies; sandwich halves together.
Notes
To make rhubarb compote, stir together 5 cups chopped rhubarb, 1¼ cups cane sugar and 1 split and scraped vanilla bean (both pod and seeds) in a pot. Cook over medium-low heat, stirring, until thickened.