Prep: 30 minutes; Rise: 1 hour 15 minutes; Cook: 8 minutes.
1. Combine yeast, 1/2 cup warm water, and honey in a small bowl or measuring cup. Let stand in a warm place for 5 minutes or until mixture begins to rise and become foamy.
2. Stir together flour, 1/4 cup sugar, salt, and cinnamon in a large bowl. Add egg and yeast mixture, and stir to blend into a sticky dough. Turn out onto a lightly floured surface, and knead until smooth and soft, about 5 minutes. Place in a greased bowl, and cover loosely; let rise in a warm place about 1 hour or until almost doubled in size.
3. Punch down dough, and roll out to 1/2-inch thickness on a lightly floured surface. Cut into 1 1/2-inch rounds, and place on a baking sheet lined with parchment or wax paper. Cover loosely, and let rise 15 to 20 minutes.
4. Heat vegetable oil in a large saucepan until a deep-fry thermometer registers 355° to 365°. Fry 4 to 5 rounds at a time until golden brown, 45 to 60 seconds per side. Drain on paper towels; roll in sugar to coat while warm. Repeat with remaining doughnuts.
5. Fill a pastry bag with jam. Make a hole in one end of each doughnut with a wooden skewer. Push pastry tip into hole, and pipe about 2 teaspoons jam into each doughnut. Serve warm.