Mix the dry. Combine 2 cups all-purpose flour, 1 ½ teaspoons active dry or instant yeast, 3 Tablespoons granulated sugar, and ½ teaspoon salt in a bowl. Make a well in the center.
Add the liquids. Stir 3 Tablespoons unsalted butter, melted, into the lukewarm ½ cup milk and whisk in 1 egg. Pour into the well and stir with a fork into a rough dough.
Knead. Knead with a dough hook for 8 minutes, or beat with a wooden spoon, until smooth and only slightly sticky.
First rise. Cover with plastic wrap. Let rise in a barely warm oven (on 30 seconds, then off) for 1½–2 hours, until doubled.
Short rest. Knead briefly on a floured surface, shape into a ball, cover with a towel, and rest 10 minutes.
Butter the dish. Grease the baking dish with some of the 4 Tablespoons unsalted butter and pour in a little more. Keep the dish warm so the butter stays liquid.
Shape the buns. Roll the dough into 1-inch (2.5 cm) ropes, cut into small pieces, and roll into balls. Coat each in the butter and arrange loosely.
Second rise. Cover and let rise 15–20 minutes. Heat the oven to 375 °F (190 °C).
Bake and brush. Bake 20–25 minutes, until golden. Brush at once with the remaining butter and cool a few minutes in the dish.
Vanilla sauce:
Warm the milk. In a saucepan, heat 1 cup milk, 1 cup half and half, 5 Tablespoons granulated sugar, and 1 teaspoon vanilla extract until warm, not hot.
Cook the sauce. Off the heat, whisk in 1 ½ Tablespoons cornstarch. Return to the heat, and once it bubbles, lower the heat and simmer 1 minute, whisking.
Add the yolks. Off the heat, let stand 2–3 minutes, then whisk in 2 egg yolks one at a time. Whisk now and then as it cools.
Serve. Pull the buns apart and pour the warm sauce over them.
Notes
Oven: Temperatures are for a conventional oven. For convection, use 350 °F (175 °C).
Liquids: Keep the milk mixture lukewarm. If it is hot, it kills the yeast.
Food safety: After the yolks go in, the sauce should read at least 160 °F (71 °C). If cooler, warm briefly without boiling.
Cream: Half-and-half is close to Czech cooking cream (12% fat). Heavy cream makes a richer, thicker sauce.
Quick version: Spread the whole dough in the dish, cut it into a grid, let it rise, and bake.
Rum: Add a splash of spiced rum to the butter for brushing.