In a large bowl or stand mixer bowl, add the tangzhong, sourdough discard, egg, egg yolk, whole milk, white sugar, instant dry yeast, and vanilla. Whisk all the wet ingredients until smooth.
All of the tangzhong, ⅓ (90g) cup sourdough discard (100% hydration), 1 egg, 1 egg yolk, ⅓ (75g) cup whole milk, ¼ (50g) cup granulated white sugar, 2 (7g) tsp instant yeast, 1½ tsp vanilla extract
Sourdough Discard Swap: If you're using extra milk and flour to substitute the discard, add the milk with the wet ingredients above and the flour with the dry ingredients below.
3 (45g) Tbsp whole milk, 6 (45g) Tbsp all-purpose flour
Add the flour and salt to the wet ingredients, and transfer to the stand mixer and knead on Speed 1 or knead by hand for about 3-4 minutes until the dough comes together.
2¾ (340g) cups all-purpose flour, 1½ (8g) tsp fine sea salt
Once the dough has come together, add the softened butter slowly by the Tablespoon. Once you've mixed in a Tablespoon, add another until all of the butter has been worked into the dough.
6 (85g) Tbsp unsalted butter
Mix the butter into the dough with the stand mixer set to Speed 1, or you can do this by kneading the butter in slowly by hand. The dough should me smooth and slightly sticky.