Combine chopped strawberries, granulated sugar, lemon juice, cardamom pods, and vanilla bean in a pot. Heat until sugar dissolves, then bring to a boil for 10 minutes. Strain solids and store the jam.

Preheat oven to 350°F. Roll out puff pastry and cut into 12 rectangles. Place half on a baking sheet and dollop jam on each. Top with remaining rectangles and seal edges.

Bake pop tarts for 15–20 minutes until golden brown. Cool on a wire rack.

Whisk together powdered sugar, milk, and jam until smooth. Adjust consistency with more milk if necessary.

Drizzle glaze over cooled tarts and let set before serving.