In the bowl of an electric mixer, whisk together the warm water, a pinch of sugar, and yeast. Let stand until mixture is foamy, about 5 minutes. Using the dough-hook attachment, mix on low speed, adding milk, butter, remaining sugar, salt, and 2 lightly beaten eggs. Gradually add enough flour to form a sticky but manageable dough. Transfer the dough to a large buttered bowl; cover tightly with buttered plastic wrap. Let dough rise in a warm spot until doubled in bulk, about 2 1/2 hours.
Line 1 large or 2 small baking sheets with parchment paper. Turn out dough onto a floured work surface. With a floured rolling pin, roll out dough to a 3/4-inch thickness. Using a 2 1/4-inch round cookie cutter, cut 24 rounds of dough as closely together as possible. Place rounds 1/4 inch apart on prepared baking sheet. Cover with buttered plastic wrap. Let rolls rise until they are light and do not spring back when pressed with your fingertip, about 30 minutes.
Preheat oven to 350 degrees. Lightly beat remaining egg with the water. Brush mixture on tops of rounds; sprinkle with poppy seeds. Bake until rolls are golden brown, 20 to 23 minutes. Transfer to a wire rack to cool for at least 5 minutes. Serve warm.
For rolls with the lightest, airiest texture, don't gather and reroll the scraps.