Preheat oven to 400 degrees F. Move baking racks to the top third of the oven and set out two large baking sheets.

Arrange the broccoli, cauliflower, carrots, and hakurei turnips on one baking sheet. Arrange the endive, radicchio, and mushrooms on another. Drizzle the vegetables with olive oil and sprinkle with sea salt and pepper.

Roast the tray with broccoli and other vegetables for 15 - 20 minutes. Roast the endive and radicchio for 10 - 15 minutes, and the mushrooms for 20 minutes. If possible, place both baking sheets on a single rack; otherwise, switch their positions halfway through cooking.

Meanwhile, make the dip. In a food processor, pulse to combine the feta, yogurt, garlic, 2 tablespoons olive oil, lemon zest and juice, sea salt, and red pepper flakes until smooth.

Arrange the vegetables on a large platter or cutting board. Spoon the dip into a bowl, drizzle with olive oil, and sprinkle with parsley, lemon zest, and red pepper flakes. Serve immediately.