How to Make It
Stir together water, yeast, and 1/2 tablespoon of the sugar in a small bowl; let stand until foamy, about 5 minutes.
Combine milk, butter, egg, salt, yeast mixture, and 2 tablespoons of the sugar in bowl of an electric mixer fitted with a dough hook. Beat on medium-low speed until well combined, about 10 seconds. With mixer running, gradually add flour, beating until flour is incorporated. Continue beating until dough is smooth and elastic, about 4 minutes. Dough will be sticky. Transfer to a medium-size lightly greased (with cooking spray) bowl; cover with plastic wrap, and let rest in a warm place (80°F to 85°F) until doubled in volume, about 1 hour.
Turn dough out onto a well-floured surface. Sprinkle top of dough lightly with flour, and pat to 1/2-inch thickness. Using a 2-inch round cutter, cut 16 circles from dough, and place on a parchment paper-lined baking sheet, leaving 1 inch between each. Discard scraps. Cover dough circles loosely with plastic wrap, and let rest in warm place until slightly increased in volume, about 30 minutes.
Heat 1 1/2-inches of oil in a Dutch oven over medium-high to 385°F. Add 5 or 6 dough rounds to oil, and fry, turning once, until golden brown on both sides, about 1 1/2 minutes total. Transfer to a rimmed baking sheet lined with paper towels; let drain 1 to 2 minutes. Place remaining 2/3 cup sugar in a large bowl; add warm doughnuts, and toss to coat. Repeat with remaining dough rounds.
Fit a piping bag with a 1/4-inch round piping tip. Poke a 1/4-inch-deep hole in top of each doughnut. Fill piping bag with strawberry jam. Insert piping tip into hole, and fill each doughnut with about 2 teaspoons jam.