Activate the yeast. Combine the warm milk, yeast, and 1 tablespoon of sugar in the bowl of a stand mixer. Stir well and let it sit for 10 minutes, until foamy. If the yeast doesn't foam your milk may have been too hot or your yeast is old. I recommend starting over with fresh ingredients.
Mix the wet ingredients. Add the eggs and yolks to the milky yeast mixture and mix on low until the eggs and yolks are broken up.
Mix the dry ingredients. Add the flour, salt, and remaining sugar to the yeast mixture and mix on low. When the dough starts to come together, add the softened butter in a couple of tablespoons at a time until all the butter is mixed in. Mix on medium-low speed for 10-15 minutes until the dough is smooth and elastic.
Let the dough rise. Transfer the dough to a greased mixing bowl and cover with plastic wrap. Let the dough rise until doubled in size, about 90 minutes. This process can take longer because of the butter, so just keep an eye on the dough and when it's doubled in size, it can be shaped.
Shape the rolls. Divide the dough into 12 equal portions (a kitchen scale works best for this), and roll them into tight balls. Place them in a greased 9x13 baking dish, cover and let the rolls proof for 1 hour, until puffed and touching.
Preheat an oven to 350 F.
Bake the rolls. Whisk together the egg wash until smooth. Brush the tops of the rolls liberally and sprinkle with flaky sea salt. Bake for 22-25 minutes, until the tops are golden and the internal temperature is 200 F.