Combine all of the tangzhong ingredients in a saucepan, whisking until no lumps remain.
Place the saucepan over medium heat, whisking or stirring constantly, until thickened, about 3 minutes.
Remove from heat and let it cool to room temperature.
Make the dough:
In a small bowl, combine the lukewarm milk (between 98 and 105 degrees F), yeast and sugar. Cover with a towel and let it stand for 5 to 10 minutes or until foamy. Reserve.
Whisk the flour and salt in the bowl of your stand mixer.
Start the mixer on low, fitted with the paddle attachment. While running, add the yeast mixture. Let it run for a few seconds before adding the tangzhong, followed by the egg and egg yolk. Mix until a shaggy dough forms.
Remove the paddle attachment and change to the hook attachment.
Start kneading and add the butter, one tablespoon at a time. Do not add the next tablespoon until the first one is fully incorporated.
Once all the butter is incorporated, let the stand mixer knead the dough for 8 to 10 minutes or until smooth and elastic.
Form a ball with the dough and place it in a greased bowl. Cover with a kitchen towel and place it in a warm spot (like your oven, turned off) for 1 to 2 hours, or until doubled in size.
Gently punch the dough to deflate it. Using a bench knife, portion the dough into 8 equal pieces, about 95g each. Form the dough portions into small balls and place them on a parchment paper-lined baking sheet, making sure they have space to double in size. Cover with the towel and let it rise again, for up to 1 hour or until puffy.
Preheat oven to 375 degrees F for at least 15 minutes.
Brush the buns with egg wash (1 large egg beaten with 1 tablespoon water), then bake for 15 to 18 minutes, or until golden brown.
Remove the buns from the oven and immediately brush them with the melted butter. Allow them to cool in the pan for 10 minutes, then transfer them to a rack to cool completely