The dressing is made with Banyuls vinegar, a French version made from sweet wine. It is mellower than red-wine vinegar.
Preheat oven to 375 degrees. Combine shallots, 2 tablespoons oil, and the vinegar in a bowl. Line a piece of foil with parchment. Place beets and 1/2 bunch thyme in center, drizzle with remaining 2 tablespoons oil, and season with salt and pepper. Fold, crimping edges to seal, place on a baking sheet, and roast until beets are tender, about 1 hour. Let beets cool slightly, then peel. Cut beets into 1/4-inch-thick slices.
Place Parmesan, cream, and remaining 2 stems thyme in a medium saucepan over medium-low heat. Gently simmer, stirring often, until cheese melts and dressing is thick, about 5 minutes. Discard thyme. Place beets on a serving platter. Drizzle with warm dressing, top with greens, then drizzle with shallot mixture. Sprinkle pistachios on top, and serve immediately.