Step 1

In a spice grinder, combine the rosemary and 1 teaspoon salt. Pulse until finely ground; transfer to a small bowl and set aside. Adjust the blade of your mandoline to slice 1/16 inch thick. If the diameter of the onion(s) is larger than the width of the mandoline blade, halve the onion(s) from end to end. Holding the whole or half onion by the root end, slice it into thin rings or half rings, slicing only as far as is safe; discard the base.

Step 2

In a medium bowl, stir together the semolina and 1 teaspoon salt. Add half of the onion slices and toss with your hands until well coated; set aside. In a large Dutch oven, heat the oil over medium-high until it reaches 350°F. Meanwhile, line a wire rack with paper towels and set it near the stove.

Step 3

When the oil comes to temperature, lift the coated onion slices out of the semolina mixture and transfer to a fine-mesh strainer. Shake to remove excess semolina, allowing the excess to fall back into the bowl. Tilt the strainer and slide the slices into the oil; cook, stirring often, until golden brown, about 3 minutes. Using a slotted spoon, transfer the fried onion to the prepared baking sheet. Immediately sprinkle with half of the rosemary salt; toss gently to coat.

Step 4

Coat, shake and fry the remaining onion slices in the same way, allowing the oil to return to 350°F before frying. Sprinkle with the remaining rosemary salt and toss to coat. Serve right away, drizzled with balsamic syrup (if using).