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  Heat milk until it's lukewarm. Stir the milk and yeast together in a large mixing bowl.
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  Add the sugar, cardamom, vanilla, and flour, stirring well. Then cut the butter into small pieces and work it into the dough. Finally add the egg and knead the dough with your hands for about 5 minutes, either in the bowl or on a floured surface. The dough will be sticky, but you can add a bit more flour if it's too sticky to knead. Cover and let rise for 30 minutes.
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  Cover a 8 x 12 inch (20 x 30 cm) pan with baking paper. Spread the dough across the pan, cover with plastic wrap or a cloth and let rise again for 20 minutes.

Topping

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  While the dough rises again, melt the butter in a saucepan over medium heat. Add the brown sugar and syrup and bring to a boil, then remove from heat and continue to stir until the mixture stops bubbling. Let the mixture cool a bit while the dough finishes rising.
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  Preheat oven to 390°F (200°C).
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  Remove the plastic or cloth from the baking pan. Press holes in the dough with your fingers, pressing down to the bottom of the pan (but not through the bottom of the dough). Pour the topping mixture across the top of the dough.
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  Bake in the middle of the oven for 15 minutes. Let the brunsviger cool a bit before slicing.