For the dough: Warm the milk and 1/2 cup water in a medium saucepan over low heat to about 100 degrees F (but no more than 110 degrees F). Remove the saucepan from the heat, and sprinkle the yeast over the liquid. Sprinkle a pinch of the sugar over the top, and let sit without stirring until foamy, about 5 minutes. Then whisk in the butter and egg yolk.
Whisk together the flour, remaining sugar and salt in a large bowl. Make a well in the center, and stir in the yeast mixture with a wooden spoon until you have a thick, sticky dough. Turn the dough onto a floured work surface, and knead until soft and elastic, about 6 minutes. Shape the dough into a ball.
Brush the inside of a large bowl with butter. Add the dough ball, turning to coat lightly. Cover the bowl with plastic wrap, trace a circle the size of the dough ball on the plastic and note the time. Let the dough rise in a warm place until doubled in size, about 1 hour 30 minutes.
Turn the dough out of the bowl, and knead briefly to release excess air. Re-form it into a ball, and return it to the bowl. Lightly butter a large piece of plastic wrap, and lay it on the dough. Cover the entire bowl tightly with the rest of the plastic, and refrigerate for 4 hours or overnight to let the dough proof.
For the filling: Melt the butter in a medium nonstick skillet over medium heat. Add the chopped shallots, and cook until soft, about 3 minutes. Add the spinach, and cook until it is cooked through and nearly all the liquid has evaporated, 5 to 7 minutes. Remove the skillet from the heat, and stir in the feta, 1/4 cup of the Parmesan, lemon zest, nutmeg, 1/2 teaspoon salt and a few grinds of pepper. Transfer the filling to a medium bowl, and let cool completely. (The filling will firm up a bit, but if it's too warm, it will be difficult to form the buns.)
To fill and form the buns: Butter a 9-by-13-inch baking pan with 2 tablespoons softened butter. Turn the dough onto a floured work surface, and press it flat. Roll it into a 10-by-18-inch rectangle, with a long side facing you. Spread the filling evenly over the dough, leaving about a 1-inch border on the side opposite you. Sprinkle the remaining 2 tablespoons Parmesan evenly over the filling. Starting from the side facing you, roll the dough up into a tight roll. Lightly brush the 1-inch border with water, and continue rolling to seal the roll. Fold the ends in just a bit, and if any spots look thicker than others, even out the roll with your hands.
Cut the roll every 1 1/2 inches with a serrated knife, to make 12 buns. Place the buns cut-side down (except for the end pieces) in the prepared pan, spacing them 1 inch apart. Cover them loosely with plastic wrap, and set in a warm place to rise until the buns double in size, leaving no gaps in between, 1 hour 30 minutes to 2 hours.
For the topping: Stir together the butter, dill, lemon zest and a pinch each of salt and pepper in a small bowl; set aside.
Position an oven rack in the center of the oven, and preheat to 350 degrees F. Bake the buns until they're golden brown and the tops spring back when pressed lightly, 25 to 30 minutes. Spread the topping evenly over the buns, and return them to the oven until the butter is melted and the topping is fragrant, 5 minutes more. Let them cool in the pan for 10 minutes before serving.
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