Heat up the milk, either in the microwave or in a saucepan, until it is "fingervarm" (finger warm), meaning it should feel warm to the touch but not hot. Measure out the sugar you will need for the dough in this recipe. If you're using active dry or fresh yeast, add the yeast along with a few teaspoons of the sugar you measured out (you can eyeball this!) into the warm milk and stir to dissolve. Let it rest 5-10 minutes until it begins to froth and bubble a little bit. (If your yeast doesn't do anything, it may be expired. Check the expiration date on the package and start over with new yeast!). If you're using instant yeast, you do not usually need to bloom it ahead of time and can add it right into a mixing bowl with all your other ingredients. If in doubt, check the package directions of the yeast you are using!
Combine the egg, egg yolk, remaining sugar, vanilla extract, and salt in a large bowl or the bowl of a stand mixer. Mix together well.
Add the milk and yeast mixture to the bowl with the rest of the ingredients and stir to combine.
Add the flour, stirring with a spoon, or using the dough hook on your mixer. Once the dough starts to come together, you can start kneading, either by hand on a clean work surface, or using the dough hook on the stand mixer (and turning up the speed a little bit).
After the flour has been added, add the softened butter and knead until smooth and elastic, probably around 5-10 minutes. Add a bit of flour if the dough sticks to your hands or the surface. If you're using a stand mixer and notice the dough sticking to the sides, be sure to add a little bit more flour as well. The texture of your dough before its first rise should be soft and smooth, not sticky and not too dry. If the dough is feeling greasy in any way from the butter, it has likely not been kneaded enough – keep kneading and adding flour if you think it needs it!
Place your dough in a bowl (or keep it in the stand mixer bowl/bowl you mixed the ingredients in), cover with a damp dishtowel or plastic wrap and let it rise in a warm place (such as a turned off oven with the light on, or a spot in your kitchen that doesn't get any drafts) for about 45 minutes to an hour or until doubled in size. You can use the "poke" test to check whether your dough is done rising; if you poke the dough and it leaves an indentation and doesn't spring back immediately, it should be ready! Depending on the conditions in your kitchen, it could take up to twice as long to rise.