In a medium bowl, combine yeast with warm water, and let stand until foamy, about 5 minutes. Whisk in honey, 1/4 teaspoon cinnamon, egg, milk, and corn oil. Using a wooden spoon, stir in salt and flour. Mix until dough appears smooth.

Transfer dough to a lightly oiled bowl, cover tightly with plastic wrap, and set in a warm place to rise until doubled in size, about 1 1/2 hours. Line a baking sheet with parchment, and another one with paper towels; set aside.

Turn out dough onto a lightly floured work surface, and knead four to five times. Roll dough into a 10-inch square, about 1/4 inch thick. Using a pizza cutter, cut into 2 1/2-by-2 1/2-inch squares, and transfer to parchment-lined baking sheet. Lightly cover with plastic; let rest in a warm place 20 minutes.

In a medium saucepan, heat peanut oil until a deep-frying thermometer registers 360 degrees. Working in batches of five or six, fry doughnuts until golden brown on both sides. Using a slotted spoon, transfer to paper-towel-lined baking sheet to drain. Fry remaining doughnuts.

In a medium bowl, combine sugar and the remaining 1/2 teaspoon cinnamon. Toss the doughnuts lightly in cinnamon sugar.

Fill a pastry bag fitted with a #4 tip with jam. Using a wooden skewer or toothpick, make a hole in the side of each doughnut. Fit the pastry tip into a hole; pipe about 1 tablespoon jam into doughnut. Repeat with remaining doughnuts. Toss filled doughnuts in cinnamon sugar again, if desired. Serve immediately.