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 35
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 35
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.
