Preheat oven or air fryer:
If baking, arrange a rack in the center of the oven and preheat to 425°F. If air frying, preheat the air fryer to 400°F.

Prepare the broccoli stems:
Using a vegetable peeler, peel the broccoli stems (or trim off the tough, fibrous outer layer with a sharp knife). Cut the stems lengthwise into 1/8″- to 1/4″-thick slices. If the stems are longer than 6″, cut them in half crosswise to yield 2 shorter pieces.

Prepare the coating:
In a medium bowl, combine the Parmesan, panko, garlic powder, salt, and paprika. Transfer about one-third of the panko mixture to a large shallow bowl. Pour the olive oil into another large shallow bowl.

Coat the broccoli:
Working a few pieces at a time, toss the broccoli stems in the oil, then dredge in the panko mixture, gently pressing to adhere. (When the panko mixture runs out, wipe out the bowl, dry it thoroughly, and replenish with another third of the mixture.) Arrange the breaded broccoli on a bare metal baking sheet.

Bake the broccoli fries:
If baking, bake on the sheet, flipping halfway through, until golden and crispy, about 10 to 12 minutes.

Air fry the broccoli fries:
If air-frying, arrange the broccoli fries in a single layer in the air fryer basket (work in batches if needed). Fry until browned, crispy on the outside, and tender in the center, about 5 minutes.

Serve:
Transfer the fries to a platter and season with additional salt. Serve with ranch or Caesar dressing on the side for dipping.