In a large skillet combine the olive oil, garlic, broccoli and salt. Over medium-low heat, gently braise the broccoli for 20 minutes. Keep the pan covered and stir well every 10 minutes.
After 20 minutes, remove the lid, turn up the heat to medium high and continue to cook the broccoli until much of the liquid is cooked off and the broccoli is charred in parts.
As the broccoli is cooking, whisk together the ingredients for the vinaigrette in a medium bowl. Taste and adjust seasoning with salt and pepper.
Transfer the broccoli to a serving dish and finish with the vinaigrette, chopped parsley and shaved Parmesan.