If using active dry yeast, heat the milk in the microwave or on the stovetop until it is “fingervarm” (finger warm), meaning it should feel warm to the touch but not hot. Add your yeast, stir, and leave it for 5-10 minutes until the mixture starts to bubble up. You can add this to the bowl of a stand mixer or another mixing bowl. If using instant yeast, simply add the yeast to the lukewarm milk – there usually is no need to wait for it to bubble up, but follow the instructions on your package of yeast if in doubt!
Add the egg, egg yolk, sugar, vanilla extract, and salt to the yeast and milk mixture.
Slowly add the flour. Once the dough starts to come together, you can begin kneading, either by hand or in a stand mixer.
Add the softened butter and knead for about 10 minutes, or until the dough is smooth and elastic. Add more flour if the dough is sticking to the sides of the mixing bowl or to your hands.
Cover the dough and let it rise for about an hour at room temperature or in a warm place, or until doubled in size.
Butter a baking sheet or a 9x13 glass pan, or line with parchment paper. 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).
While the oven preheats, 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 slightly (but not harden or congeal!).
Once the dough has risen again, poke holes in it with your fingers. Then, add the topping, and if the topping is cool enough, you can use your fingers to poke more holes. You want to push pretty far down, almost to the bottom of the baking tray.
Bake the brunsviger for 15-20 minutes, or until the top of the dough looks nice and golden brown. Depending on the pan you've used and the thickness of the dough (thin dough on a dark metal baking sheet will bake faster), you may want to check the doneness as early as 10 minutes! Enjoy!