Make dough: In a medium saucepan, heat milk and water over medium heat until scalding. (An instant-read thermometer should register 180°F/82°C.) Remove from heat and add butter; stir occasionally until butter is melted, and an instant-read thermometer registers 120°F (49°C) to 130°F (54°C).
In the bowl of a stand mixer fitted with the paddle attachment, beat 2 cups (250 grams) flour, sugar, poppy seeds, salt, yeast, and lemon zest at low speed until combined. Add warm milk mixture and beat until combined. Add egg, beating until combined. With mixer on low speed, gradually add 2 cups (250 grams) flour, beating just until a shaggy dough comes together and stopping to scrape sides of bowl.
Switch to the dough hook attachment. Beat at low speed until a soft, somewhat sticky dough forms, about 24 minutes, stopping to scrape sides of bowl and dough hook; add up to remaining ¼ cup (31 grams) flour, 1 tablespoon (8 grams) at a time, if dough is too sticky. Turn out dough onto a clean surface; knead 5 to 6 times, and shape into a smooth round.
First rise: Lightly oil a large bowl. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, about 1 hour.
Cut 8 (5-inch) squares of parchment paper, and place on a rimmed baking sheet.
Shape doughnuts: Turn out dough onto a lightly floured surface and roll to ½-inch thickness. Using a 3½-inch doughnut cutter or a 3½-inch round cutter dipped in flour, cut dough. Gently transfer doughnuts to prepared parchment squares, spacing at least 2 inches apart.
Reroll scraps; let dough stand for 10 minutes. Cut dough, and place on prepared parchment squares. (Discard any remaining dough scraps or roll into small balls for doughnut holes.)
Second rise: Cover with plastic wrap, and let rise in a warm, draft-free place (75°F/24°C) until puffed, about 1 hour.
Prepare oil: In a 4- to 6-quart cast-iron Dutch oven or other heavy-bottomed saucepan, pour oil to a depth of 2 inches, and heat over medium heat until a deep-fry thermometer registers 365°F (185°C).
Line a rimmed baking sheet with paper towels.
Fry doughnuts: Using parchment paper to pick up doughnuts, transfer doughnuts in batches to hot oil, letting doughnuts gently fall off parchment into oil. (Do not place parchment paper in oil.) Fry doughnuts in batches until golden brown, 1 to 2 minutes per side. Using a spider strainer, remove doughnuts, and let drain on prepared pans. (See note on how to bake these doughnuts.)
Make Lemon Glaze: (makes about ¾ cup) In a medium bowl, whisk together confectioners’ sugar, lemon juice, water, and vanilla. Use immediately.
Glaze as desired with Lemon Glaze. Best served same day.