Step 1

Add the yeast, granulated sugar, salt and 3 cups of the flour to the bowl of a stand mixer. Whisk to combine and to break up any lumps of flour. Add the water and use a spatula to begin to mix the ingredients. Add the melted butter and the remaining 3 cups of flour and mix again until a very shaggy dough forms. Attach the bowl to the mixer fitted with the dough hook. Knead the dough on medium-low until all the dough has pulled away from the sides of the bowl and is elastic and soft to the touch, about 7 minutes.

Step 2

Remove the dough from the mixer and divide into 2 pieces. Add one half back to the mixer bowl and add the other half to a large glass bowl. Pour 1 tablespoon of olive oil over the dough in the glass bowl; turn it once to coat and cover it with plastic wrap or a clean kitchen towel.

Step 3

Add the rosemary to the dough in the mixer. Turn the mixer on and knead until the rosemary is evenly incorporated, 2 to 3 minutes. Remove the dough from the mixer and place it in a separate large glass bowl. Drizzle it with the remaining tablespoon of olive oil; turn it once to coat and cover it with plastic wrap or a clean kitchen towel. Allow both doughs to proof in a warm place until doubled in size, about 1 hour.

Step 4

After the dough has doubled in size, fill a large Dutch oven, fitted with a deep-fry thermometer, with about 2 inches of vegetable oil and heat to 360 degrees F. Line two rimmed baking sheets with parchment paper and two rimmed baking sheets with wire racks. Set aside.

Step 5

To assemble the Parmesan-rosemary dough balls, use your thumb and index finger to pinch off about 2 tablespoons of dough. Place a cube of Parmesan in the center and close the dough around it, pinching the seam to seal it. Place it seam side down on the parchment-lined baking sheet. Continue the process with the remaining rosemary dough and Parmesan. Set aside the finished dough balls.

Step 6

To assemble the jam-filled dough balls, use your thumb and index finger to pinch off 2 to 3 tablespoons of dough. Flatten the dough slightly, then spoon about 1/2 teaspoon of the jam into the center of the dough. Carefully pull the edges of dough up and over the jam, pinching the seam to seal it, making sure to leave no jam exposed. Place it seam side down on the second parchment-lined baking sheet. Continue the process with the remaining dough and jam.

Step 7

When the oil is at 360 degrees F, carefully lower 5 to 6 Parmesan-rosemary dough balls into the hot oil. Fry, turning as needed until golden brown on all sides, 2 to 3 minutes. Remove to a wire rack-lined baking sheet. Immediately sprinkle with salt. Fry the remaining Parmesan-rosemary dough balls, allowing the oil to come back up to temperature before adding another batch. Serve immediately or keep warm in a low oven for 20 to 30 minutes. Top with freshly grated Parmesan before serving.

Step 8

To fry the jam-filled dough, carefully lower 5 to 6 dough balls into the hot oil Fry, turning as needed until golden brown on all sides, 2 to 3 minutes. Remove to the second wire rack-lined baking sheet. Fry the remaining jam-filled dough balls, allowing the oil to come back up to temperature before adding another batch. Serve immediately or keep warm in a low oven for 20 to 30 minutes. Top with confectioners' sugar before serving.