In a white ceramic mixing bowl, whisk together 1 cup warm milk (about 110°F), 1 tablespoon of the sugar, and the 2 1/4 teaspoons active dry yeast. Let the mixture stand for 5 to 10 minutes, until it looks foamy on top, which is how you know the yeast is alive.

Add the remaining sugar, 1 large egg, 1 large egg yolk, 3 tablespoons melted butter, 1 teaspoon vanilla extract, and 1 teaspoon salt to the bowl. Stir in 3 1/2 cups flour, about a cup at a time, until a shaggy, sticky dough comes together.

Turn the dough onto a lightly floured pale wood board and knead by hand for 8 to 10 minutes, or with a stand mixer's dough hook on medium-low for 6 minutes, until it turns smooth, elastic, and only slightly tacky. Add flour a tablespoon at a time, only if the dough sticks heavily to your palms.

Shape the dough into a ball, return it to the white ceramic mixing bowl, and cover with a clean kitchen towel. Let it rise in a warm spot for 1 to 1 1/2 hours, until it has doubled in size and holds a slight dent when you press two fingers into it.

Punch the dough down gently to release the air, then turn it out onto the floured pale wood board. Roll it out with a rolling pin to an even 1/2 inch thickness.

Cut rounds with a 3-inch round cutter, pressing straight down without twisting the cutter. Gather the scraps, re-roll once, and cut more rounds until you have about 14 rounds total.

Arrange the rounds on a parchment-lined baking sheet, spacing them an inch apart, and cover loosely with a towel. Let them proof for 45 minutes to 1 hour, until visibly puffy; they should feel soft and pillowy, not dense, when you touch one lightly.

Pour about 3 inches of vegetable oil into the cream enameled Dutch oven and heat over medium heat to 350°F, checking with a deep-fry thermometer clipped to the side of the pot. Hold the oil between 345°F and 355°F for the whole frying stretch.

Lower 3 to 4 dough rounds into the hot oil at a time, leaving room for them to float freely. Fry for 1 1/2 to 2 minutes per side, turning once with a slotted spoon, until deep golden brown all over and puffed through the middle.

Lift the doughnuts out with the slotted spoon and set them on a wire rack over paper towels to drain. Let them cool completely at room temperature for at least 20 minutes before filling; a warm center melts the jam and pushes it straight back out.

Fit a piping bag with a long, narrow bismarck tip and fill it with 1 cup strawberry jam. Insert the tip about an inch into the side of each cooled doughnut and squeeze gently until you feel slight resistance and see a little jam at the opening, about 1 tablespoon per doughnut.

Spoon 1 cup powdered sugar into a small fine-mesh sieve and dust it generously over each filled doughnut, turning the doughnut to coat the top evenly.

Pile the finished sufganiyot onto a white ceramic platter and serve them the same day, while the outside still holds a little crispness against the soft middle.