Heat up the milk until it is "fingervarm" (finger warm), meaning it should feel warm to the touch but not hot. Add the yeast and stir to dissolve. Let it rest 5-10 minutes until it begins to bloom and froth a little bit.
Meanwhile, combine the egg, egg yolk, sugar, vanilla extract, and salt in a large mixing bowl and mix well.
Once the yeast has bloomed, add the yeast mixture to the egg mixture and whisk together.
Slowly add the flour. Once the dough starts to come together, you can begin kneading.
Add the unsalted butter and knead for about 10 minutes, or until the dough is smooth and elastic. Add more flour if the dough is sticking to your hands or the table.
Cover the dough and let it rise for about an hour at room temperature.
Butter a baking tray. Once the dough has doubled in size, you can either roll it out or just stretch it with your hands to fit the tray.
Poke holes in it with your fingers. Let this rise another 30-45 mins, and preheat your oven to 400 degrees F (200 C).
In the meantime, make your topping. Combine the brown sugar and unsalted butter in a saucepan and place it over a low heat. You want the sugar to dissolve and the butter to melt, but if you rush it at a higher heat, they will separate. If that happens, add a little bit of hot water and stir to try to save the mixture. Let your mixture cool a little bit (but not harden or congeal!).
Once the dough has risen again, add the (slightly cooled) topping, and use your fingers to poke more holes. You can also make the holes first and then pour the topping over that! You want to push pretty far down, almost to the bottom of the baking tray.
Bake the brunsviger for 18-20 minutes, or until the top of the dough looks nice and golden brown. If you bake in a convection oven and use a glass pan, you may want to check the doneness as early as 15 minutes! Enjoy!