Bring milk to a simmer in a 1-quart heavy saucepan, then remove from heat and stir in sugar and salt. Pour into a large bowl and let cool slightly.

While milk is cooling, dissolve yeast in warm water in a small bowl, stirring. Let stand for 5 minutes.

Stir the yeast mixture into the milk, along with 2 tablespoons oil, eggs, and zest. Stir in 2 1/2 cups flour with a wooden spoon to make a very soft dough. Spread 1 cup flour on work surface and put dough on top, scraping it from bowl with a rubber spatula. Knead dough, incorporating all of flour from work surface and adding just enough additional flour (if necessary), about 5 minutes; dough should be slightly sticky. Transfer dough to an oiled large bowl and sprinkle lightly with additional flour. Cover bowl with a clean kitchen towel and let dough rise in a warm draft-free place until doubled in bulk, about 1 1/2 hours.

Turn out dough onto a floured surface and roll out with a floured rolling pin until 1-inch thick. Cut out rounds with a 2-inch cutter. Place on a sheet pan and cover doughnuts with another kitchen towel and let rise in a warm place for 30 minutes.

While doughnuts rise, heat oil in a deep 4-quart pot until it registers 365 degrees F on a thermometer. Fry doughnuts 2 at a time, turning once or twice, until puffed and golden brown, about 2 to 3 minutes per batch. Transfer to paper towels to drain.

Place jam in a pastry bag fitted with a 1-inch plain tip and poke the tip into the center of each doughnut (entering from the side) and squeeze in some of the jam. Ice the top with some of the ganache and sprinkle with hazelnuts.