Make the doughnuts: Put 1/4 cup warm water (110 degrees F to 115 degrees F) in a stand mixer bowl. Add a pinch of granulated sugar, then sprinkle the yeast on top; let stand until foamy, about 10 minutes.
Whisk 1/2 cup granulated sugar, the milk, eggs, 4 tablespoons melted butter, the lemon zest, salt and vanilla in a separate bowl. Add to the yeast mixture and whisk to combine. Using the dough hook on medium speed, gradually beat in the flour, scraping down the bowl as needed. Continue beating until the dough gathers around the hook, about 3 minutes. Turn out onto a clean surface and shape into a ball.
Brush a large bowl with butter. Add the dough, turning to coat. Cover with plastic wrap and let rise in a warm place until doubled in size, 1 hour to 1 hour 30 minutes. (If making the dough ahead, punch it down, wrap in plastic and refrigerate overnight; bring to room temperature before continuing.)
Turn the dough out onto a lightly floured surface; roll out into a 12-inch round, about 1/2 inch thick. Cut out 12 circles with a floured 3-inch-round cookie cutter, then cut out the centers with a 3/4-inch-round cutter. Line a baking sheet with parchment paper and lightly coat with cooking spray. Arrange the doughnuts and doughnut holes on the baking sheet, leaving space between each; cover loosely with plastic wrap and set aside at room temperature until slightly puffed, about 30 minutes.
Heat 2 inches of vegetable oil in a large pot over medium-high heat until a deep-fry thermometer registers 340 degrees F. Use scissors to cut the parchment around each doughnut, leaving a 1-inch border. Carefully invert 2 doughnuts into the oil and peel off the parchment with tongs. Fry until golden, 1 minute 30 seconds to 2 minutes per side. (Fry the doughnut holes 1 minute per side.) Remove with a slotted spoon and drain on paper towels. Bring the oil back to 340 degrees F and repeat with the remaining doughnuts.
Make the glaze: Whisk the confectioners' sugar, milk, lemon zest, vanilla and salt in a bowl. Dip the warm doughnuts in the glaze.