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PreparationFor the Dough: Heat milk in a small saucepan over low until just warm; an instant-read thermometer should register 105°F–115°F. Pour into the large bowl of a stand mixer. Whisk in yeast and let sit until foamy, 10–15 minutes. You should see a layer of foam on the surface; this means that the yeast is active.
Add egg, egg yolk, flour, granulated sugar, cardamom, and salt to yeast mixture and mix with dough hook on low speed until well combined. Increase speed to medium and continue to mix until dough is smooth and elastic, about 10 minutes. Gradually add butter 1 Tbsp. at a time, mixing to incorporate slightly before adding more. When 6 Tbsp. butter are incorporated, mix until dough is smooth, supple, and shiny, 10–15 minutes longer (you need to mix this long to develop the gluten).
Transfer dough to a large buttered bowl, cover with plastic wrap and a kitchen towel, and let sit at room temperature until doubled in size, 1–1 1/2 hours. Punch dough down, cover bowl tightly in plastic wrap, and chill overnight (the slow proof creates more flavor because it gives the yeast more time to process the sugar).
For the Filling and Assembly: Mix butter, cinnamon, and 1/2 cup brown sugar in a medium bowl until combined.
Have your tools and filling nearby; you want to work quickly and keep dough as chilled as possible to make rolling more manageable. Turn out dough onto a lightly floured piece of parchment paper and roll to a 1/4"-thick rectangle, about 16"x12". Cover with plastic and chill until dough is firm, 30–60 minutes.
Spread butter mixture over two-thirds of dough. Fold plain side over the middle, then fold opposite third over (like folding an envelope). Roll to a 12"x8" rectangle about 1/2" thick. Slice lengthwise into eight 1"-thick strips.
Working one at a time, lay strip on clean work surface with long side facing you. Using your palms, gently twist each end in opposite directions until entire strip is spiraled. Hold one end of strip between a thumb and index finger. Working away from you, tightly wrap strip around 3 fingers. Wrap around fingers again, placing second loop closer to the palm of your hand. As you bring strip across the back of your hand, cross over first loop, angling toward the end of your index finger. Cross dough over front of your hand and tuck the end into the center while removing your fingers and pushing through to the other side to create a knot. Check out step-by-step photos here. You want to twist these tightly so that they rise while baking. If you’re having trouble, twist them into any kind of knot that you can—just make sure it's wrapped tightly.
Divide cinnamon twists between 2 rimmed baking sheets, spacing evenly apart. Cover loosely with plastic wrap and let sit at room temperature until doubled in size, 45–60 minutes.
Place racks in upper and lower thirds of oven; preheat to 325°F. Bake cinnamon buns, rotating pans from top to bottom halfway through, until deeply golden brown, 25–30 minutes.
Bring remaining 1 cup brown sugar and 1/2 cup water to a boil in a small saucepan. Generously brush syrup over hot buns. If you like a sweeter pastry, wait 5 minutes, then brush again.
Do Ahead: Buns can be made 2 days ahead. Tightly wrap with plastic and store at room temperature.