Combine yeast, 1/2 cup warm (110°) water, and 1 tsp. sugar in a small bowl. Set aside until foamy, about 5 minutes.
Put 2 3/4 cups flour in bowl of a stand mixer or mixing bowl and make a well in the center. Add eggs, yeast mixture, 1/4 cup sugar, the butter, nutmeg, and salt. Mix with dough hook on medium speed, scraping inside of bowl occasionally, until dough is smooth and stretchy, 8 to 10 minutes. If needed, add about 1/4 cup more flour and mix until dough pulls free from bowl and is no longer sticky. (To mix by hand, stir with a spoon, then knead on a board until smooth and stretchy, 8 to 10 minutes, adding flour to keep dough from sticking.)
Rub dough with a bit of oil, turn over in bowl, and cover. Let rise in a warm place until doubled, 1 hour. In a bowl, mix remaining sugar and the cinnamon.
Push dough down to release air and shape into a smooth ball. Roll on a lightly floured surface to 1/4 in. thick. Using a 2 3/4-in. round cutter or drinking glass, cut rounds and place on 2 lightly oiled baking sheets; gather scraps and reroll as needed. Cover lightly with plastic wrap and let rise in a warm place until puffy, 20 minutes.
Pour 1 in. oil into a 5- to 6-qt. pan, insert a deep-fry thermometer, and warm oil over medium-high heat to 325° to 350°. Using a slotted spoon, slip 4 rounds into oil, one by one. Fry until deep golden, turning once, 2 to 3 minutes total; adjust heat as needed to keep oil between 275° and 350°.
Lift out doughnuts with slotted spoon, letting drain slightly, and put in bowl of cinnamon sugar. Turn in sugar while warm, using a fork, then set on a platter. Repeat with remaining rounds.
Poke each doughnut to center with a teaspoon handle. Fit a pastry bag with a 1/3-in.-wide round tip (or snip a corner off a resealable plastic freezer bag), then fill with jam. Pipe or squeeze about 2 tsp. jam into each doughnut (you may need to squeeze doughnuts to hold them open).
Make ahead: Up to 4 hours, wrapped airtight, at room temperature.
Note: Nutritional analysis is per doughnut.
Servings Makes about 20
* Percent Daily Values are based on a 2,000 calorie diet. Your daily value may be higher or lower depending on your calorie needs.