Peel the hard, woody exteriors of the broccoli stems. Cut each stem in half lengthwise, and then again into quarters.
Toss the broccoli stem fries with olive oil in a bowl, then coat in Parm. Line a baking sheet with parchment paper and spray lightly with cooking spray. Place the fries on the parchment. Bake until golden brown, 15 to 20 minutes. Sprinkle with flakey salt and serve immediately.