Preheat the oven to 450°F (235°C) and set an oven rack in the middle position. Line a baking sheet with aluminum foil.

Cut the broccoli into 2-inch (5-cm) florets, leaving 1 to 2 inches (2.5 to 5 cm) of stems attached. Discard the remaining thick stalks. In a large bowl, combine the olive oil, garlic, salt, pepper, and sugar and mix well. Add the broccoli and toss to coat evenly. Transfer the broccoli to the prepared baking sheet.

Roast for 10 minutes, until the broccoli is crisp-tender and lightly browned in spots. Sprinkle 2 tablespoons of the grated Parmigiano Reggiano over the broccoli. Taste and adjust seasoning if necessary.

Transfer to a serving dish and gently squeeze 1 to 2 teaspoons of lemon over the top. Sprinkle with the remaining cheese and serve.