In a small bowl, stir together the lukewarm water, yeast, and 1 tablespoon sugar. Let sit for 10 minutes, until foamy and active on top. If it does not foam, start again.
In the bowl of a stand mixer fitted with the dough hook, combine the flour and 50 grams sugar. Add the whisked eggs, melted butter, room temperature milk, and the yeast mixture.
Mix on low until a shaggy dough forms, then increase to medium-low and knead for about 10 minutes, until the dough becomes smooth, elastic, and slightly tacky but not sticky. It should pull away from the sides of the bowl.
Transfer the dough to a lightly oiled or floured bowl, cover, and let rest at room temperature until doubled in size, about 1 to 1 1/2 hours depending on your kitchen.
Turn the dough out onto a lightly floured surface and roll into a rectangle about 1/2 inch thick. Using a 3 1/2-inch round cutter, cut out circles. Use a smaller cutter to cut out the center holes.
Place each doughnut onto its own small square of parchment paper. This makes transferring them into the oil much easier later and keeps their shape intact.
Cover loosely and let rest again for 45 minutes, until noticeably puffy.
Meanwhile, heat neutral oil in a heavy-bottomed pot to 335 to 350°F. Keep the oil within this range. Lower than 335°F and they will absorb oil. Higher than 350°F and they will brown too quickly before cooking through.
Mix together your glaze ingredients, and set to the side.
Carefully lower the doughnuts into the oil, using the parchment to guide them in. Fry for 1 to 2 minutes per side, until deeply golden and puffed. Do not overcrowd the pot. I prefer to fry two at a time.
Set onto a paper towel lined baking sheet, just to drain of any excess oil.
Dip into your glaze and set onto a wire rack placed inside a baking sheet lined with paper towels. Let set for a few minutes before serving so the excess glaze has time to drip off. Best enjoyed the same day !!