In a medium bowl, stir together the yeast and the water. Set aside to "proof".
After 10 minutes, stir in the 1 cup of all-purpose flour, cover with plastic wrap and set aside in a warm place.
In the bowl of an electric mixer fitted with a paddle attachment, cream together the butter and sugar for 8 to 10 minutes. With the mixer on low speed, add the eggs, 1 at a time. When the mixture is smooth, stir in the vanilla and salt. Add the bread flour and mix on low speed until fully incorporated.
Stir in the yeast mixture until thoroughly incorporated. Put the dough into an oiled bowl and allow to proof for 1 to 1 1/2 hours. In a medium bowl, mix together the granulated sugar and the ground cinnamon. Set aside.
Lightly flour a cool work surface and roll the dough out to about 1 1/2-inch thickness. Cut the dough with a round 3-inch cutter and turn them onto a floured baking sheet to rest, about 20 to 30 minutes, before frying.
Heat the oil in a large heavy-bottomed pot to 365 to 370 degrees F. Drop the dough rounds, in batches, into the hot oil and cook until golden brown on 1 side. Using a slotted spoon, turn the dough rounds on the second side and cook until golden brown. Remove them from the oil to paper towels to remove any excess oil. Roll the doughnuts in the cinnamon sugar and arrange on a serving platter. Serve immediately with the Cranberry Jam for dipping.
In a medium pot, add the cranberries, sugar, water and the 1/2 cup of orange juice. Stir to blend and bring the mixture to a boil over medium heat. Reduce the heat and add the cinnamon stick, allspice and nutmeg. Stir to blend and simmer for an additional 5 to 7 minutes. Some of the cranberries will burst and some will remain whole. Stir in the orange zest and orange juice and transfer to a serving bowl. Serve warm with the doughnuts.
Recipe courtesy of Alex Guarnaschelli