In a small bowl, mix the warm milk, 1 tablespoon of sugar, and the yeast. Let it sit for about 5–10 minutes until it becomes frothy.
In a large mixing bowl or the bowl of a stand mixer fitted with a dough hook, combine the flour, remaining sugar, and salt.
Add the yeast mixture and eggs to the dry ingredients. Mix the dough in the stand mixer or by hand until it becomes smooth and elastic, about 5–10 minutes.
Gradually add in the butter, a few pieces at a time, while continuing to knead. Make sure each batch of butter is fully incorporated before adding the next. The dough should become smooth and slightly sticky. This may take up to 10 minutes.
Place the dough in a greased bowl and cover it with a damp cloth or plastic wrap. Let it rise in a warm place until it has doubled in size, about 1–2 hours.
After the dough has risen, punch it down gently. Divide the dough and shape it into loaves or rolls, depending on your preference.
Place the shaped dough into greased loaf pans or on a baking sheet if making rolls. Cover and let rise until nearly doubled, about 30–60 minutes.
While the dough is rising, preheat your oven to 375°F (190°C). Prepare the egg wash by beating together 1 egg with 1 tablespoon of milk or water. Gently brush the risen dough with the egg wash.
Bake the loaves for about 25–30 minutes or rolls for about 18–20 minutes, or until the bread has a golden-brown crust and sounds hollow when tapped on the bottom.
Remove the bread from the oven and let it cool in the pans for a few minutes before transferring to a wire rack to cool completely.