Step 1: Preheat your oven to 400°F (200°C). Prepare beets and carrots. For beets, scrub well and trim ends, leaving skin on or peeling. Cut larger beets into quarters, smaller ones in half. For carrots, peel and slice into 1/2-inch thick rounds or diagonal pieces.
Step 2: In separate bowls, toss beets with 2 tablespoons of olive oil, salt, and pepper. Do the same with carrots using 1 tablespoon of olive oil, salt, and pepper. Spread them in a single layer on a baking sheet (separate if preferred to avoid beet stain). Roast for 25-40 minutes, or until tender and slightly caramelized, tossing halfway. Let cool slightly, then peel beets if unpeeled, and cut to desired size.
Step 3: While vegetables cool, prepare the dressing. In a small bowl, whisk together the remaining 2 tablespoons of olive oil, fresh lemon juice, Dijon mustard, minced garlic, chopped fresh dill, and chopped fresh parsley. Season with salt and pepper to taste.
Step 4: Arrange the fresh arugula on a large serving platter or individual plates. Scatter the cooled roasted beets and carrots over the arugula.
Step 5: Place the burrata cheese on top of the vegetables. Gently tear or slice the burrata just before serving to expose its creamy interior.
Step 6: Drizzle the lemon-herb dressing generously over the entire salad. Garnish with extra fresh dill and parsley, and a final grind of black pepper. Serve immediately.