In a small bowl, stir together warm water, yeast, and 1/2 tablespoon of sugar. Let stand until foamy (about 5 minutes).

In the bowl of an electric mixer fitted with a dough hook, combine milk, melted butter, beaten egg, salt, yeast mixture, and 2 tablespoons of sugar.

Beat on medium-low speed until well combined (about 10 seconds).

With the mixer running, gradually add flour, beating until incorporated.

Continue beating until the dough is smooth and elastic (about 4 minutes). Note: Dough will be sticky.

Transfer the dough to a lightly greased bowl (use cooking spray).

Cover with plastic wrap and let rest in a warm place (80°F to 85°F) until doubled in volume (about 1 hour).

Turn the dough out onto a well-floured surface. Sprinkle the top lightly with flour.

Pat the dough to 1/2-inch thickness.

Using a 2-inch round cutter, cut 16 circles from the dough. Place them on a parchment paper-lined baking sheet, leaving 1 inch between each.

Discard the scraps.

Cover the dough circles loosely with plastic wrap and let rest in a warm place until slightly increased in volume (about 30 minutes).

Heat 1 1/2 inches of oil in a Dutch oven over medium-high heat to 385°F.

Add 5 or 6 dough rounds to the oil and fry, turning once, until golden brown on both sides (about 1 1/2 minutes total).

Transfer the fried doughnuts to a rimmed baking sheet lined with paper towels to drain for 1 to 2 minutes.

Place the remaining 2/3 cup of sugar in a large bowl. Add the warm doughnuts and toss to coat. Repeat with the remaining dough rounds.

Fit a piping bag with a 1/4-inch round piping tip. Poke a 1/4-inch-deep hole in the top of each doughnut.

Fill the piping bag with strawberry jam. Insert the piping tip into the hole and fill each doughnut with about 2 teaspoons of jam.