Cut 1 head broccoli into florets. Peel the stem, shave into ribbons and soak in ice water. Heat 2 1/2 tablespoons olive oil in a skillet over medium heat. Add the broccoli florets; cook, stirring occasionally, until charred, about 6 minutes. Transfer to a bowl. Add 1 tablespoon capers to the skillet; cook 30 seconds, then add to the broccoli. Drain the shaved broccoli stem; add to the bowl with the florets and toss with 1 tablespoon each lemon juice and olive oil, 2 tablespoons chopped parsley, 1/3 cup grated parmesan, and salt and pepper.