To the bowl of a stand mixer, combine the tangzhong mixture, milk, sugar, salt, instant yeast, 2 eggs and about half of the flour. On low speed, mix with the dough hook attachment until combined and smooth. Gradually add the remaining half of the flour. Continue to mix until fully combined. The dough will be thick and slightly dense.
While on low speed, add the softened butter, one tablespoon at a time. Allow butter to fully incorporate between additions. This could take up to 5 minutes. While adding the butter, the dough will begin to smooth out and become softer.
Knead dough for 10-15 minutes on low speed, until it forms a cohesive, smooth ball of dough. It should come to window pane (check blog post for tips on the windowpane test). At this point, the dough should feel very soft and elastic. It will become easier to handle after it rises during bulk fermentation.