Position racks in the upper and lower thirds of the oven; preheat to 350 degrees F. Line 2 baking sheets with parchment paper.
Make the twists: On a floured surface, roll out the dough into a 10-by-18-inch rectangle. Spread the poppy seed filling evenly over the dough, leaving a 1-inch border on one of the long sides. Brush the clean border with water. Tightly roll the dough into an 18-inch log, rolling toward the clean border.
Using a chef's knife, cut the log crosswise into 12 rounds, 1 1/2 inches wide each. Dampen your hands, then stretch and pat each round into an oval shape. Twist each oval in the middle to form a figure-eight shape. Place on the prepared baking sheets. Cover loosely with plastic wrap and set aside in a warm place until plump, 20 minutes.
Uncover the dough and retwist, if necessary. Bake until golden, about 20 minutes, switching the position of the baking sheets halfway through. Transfer to racks and let cool slightly on the baking sheets.
Make the glaze: Combine the confectioners' sugar, butter, lemon juice and zest, milk, vanilla and salt in a microwave-safe bowl. Cover with plastic wrap and microwave 45 seconds; stir until smooth. Brush on the twists.
Warm 1/2 cup water and the milk in a saucepan over low heat until a thermometer registers 100 degrees F to 110 degrees F. Remove from the heat and sprinkle the yeast on top, then sprinkle with a pinch of the sugar; set aside, undisturbed, until foamy, about 5 minutes.
Whisk the melted butter, egg yolk and vanilla into the yeast mixture until combined. In a large bowl, whisk the flour, the remaining sugar, the salt and nutmeg. Make a well in the center, then add the yeast mixture and stir with a wooden spoon to make a thick and slightly sticky dough. Turn out onto a floured surface and knead until soft and elastic, about 6 minutes. Shape into a ball.
Brush a large bowl with butter. Add the dough, turning to coat lightly with the butter. Cover with plastic wrap and let rise at room temperature until the dough is doubled in size, about 1 hour, 15 minutes.
Turn the dough out of the bowl and knead briefly to release excess air; re-form into a ball and return to the bowl. Lightly butter a large piece of plastic wrap and lay it directly on the surface of the dough. Cover the bowl tightly with plastic wrap and refrigerate at least 4 hours or overnight.