Add the butter to a small saucepan and set it over medium heat. Stir the butter occasionally until it starts to foam and turn amber in color, then remove it from the heat.
Transfer the butter to a heat-safe container and weigh it -- you should have about 113g left. If you have a lot more, you'll want to return it to the heat and continue cooking. If you have a lot less, you've unfortunately burned the butter and will want to start the process again.
Set the butter aside at room temperature and start working on the dough.
Tangzhong
In a small saucepan over medium heat, continuously stir together the flour and milk until the mixture and thickens into a thick slurry. Take off the heat and set aside. Alternatively, whisk together the flour and milk in a microwave safe bowl. Microwave in 30 second intervals, stirring after each, until the mixture becomes a thick slurry.
Add the flour, sugar, yeast, salt, 42g of brown butter, egg, milk, and tangzhong mixture to the bowl of a stand mixer fitted with the dough hook attachment.
Knead the mixture on medium speed for ~15-20 minutes. The dough is ready when it is no longer sticking to the sides or bottom of the bowl, and passes the windowpane test.*
Once the dough is done kneading, shape the dough into a ball and place it back into the mixing bowl. Cover the mixing bowl with plastic wrap, and place it in a warm area to rise until doubled in size (~1 hr.)**
Brown Butter Garlic Filling
Add 42g of brown butter to a small mixing bowl, melting it if it has firmed up.
Use a fine grater to grate the garlic cloves into the butter, then add in a pinch of salt, a generous amount of black pepper, and some red pepper flakes if you want a little spice in the rolls. Mix well.
Once the dough has finished rising, lightly punch it down to remove any excess air in it. Transfer the dough to a clean and lightly floured surface, then roll it out to a 12″ x 16″ (30 x 40 cm) rectangle. If the dough keeps springing back a lot while rolling it out, let it rest for 5 minutes before rolling it again.
Pour the butter filling on to the dough, then gently spread it out into an even layer.
From the longer side, fold the dough in half. Cut the dough into 6 strips that are ~2" wide and ~8" long.
Cut each strip in half long-ways, leaving the strip still attached at the top. Twist the two strips together, then make a circle with the twisted dough and bring one end of the strip through the circle and gently pull on it to make a knot.
Transfer the rolls to a lightly buttered and/or lined eighth sheet pan*** Lightly cover the pan with a towel, and let the rolls rise in a warm environment until doubled in size (~30 minutes).
Egg Wash
Preheat the oven to 375°F / 190°C.
Make an egg wash by whisking together an egg with 30g of milk, then brush it over the roll right before baking. (You will not use all of the egg wash, store the rest in the fridge or freezer for other pastries!)
Bake for 18-22 minutes, until the tops of the knots are golden brown and the internal temperature of the rolls has reached 190°F / 88°C.
Topping
Melt the remaining brown butter and brush it over the rolls while they are still warm. Top the rolls with freshly grated parmesan and sliced green onions, then serve & enjoy!
Notes
Affiliate links: Kitchen Scale (my favorite) | Kitchen scale (great starter option) | Oven Thermometer | Cordless Hand Mixer | Stand Mixer| Eighth Sheet Pan*Take a small piece of dough and stretch it gently as far as you can. If it tears super easily, keep kneading. If it's able to spread to the point where it's super thin and you can see light through it, it has passed the windowpane test and is good to go!**At this point you could leave the dough in the fridge overnight, and proceed with the rest of the recipe the next day.***An eighth sheet pan is 10×7″. You could also use an 8″ circle pan.