Whisk butter until creamy.
Combine flour, water, and salt; knead until tacky.
Roll dough into a ½-inch-thick rectangle, rest for 15 minutes.
Roll into a 15-inch square, spread a third of the butter, fold, and seal.
Repeat with remaining butter, rolling into an 18-inch square.
Roll into a log, wrap, and refrigerate for at least 2 hours.
Boil sugar, water, and cinnamon stick for 1 minute, then let sit.
Heat 1 cup plus 1 tablespoon milk until bubbles form.
Whisk flour with remaining milk, then slowly add hot milk.
Discard cinnamon stick, add sugar syrup, and cook until thickened.
Whisk in egg yolks, strain mixture.
Preheat oven to 260°C.
Cut dough log into ½-inch slices, press into tart molds.
Fill each shell with 1½ tablespoons of filling.
Bake until golden brown and caramelized, about 15-20 minutes.
