In a sauté pan over medium-high heat, warm the olive oil. Add the butternut squash and cook, stirring occasionally, until lightly browned and just tender, 7 to 9 minutes. Season with salt and black pepper. Add the garlic and red pepper flakes and quickly toss with the squash.
Add the zucchini and cook, stirring occasionally, until just tender, about 2 minutes. Season with salt and black pepper.
Remove the pan from the heat and stir in the grated cheese. Transfer to individual bowls. Garnish with shaved cheese and drizzle with lemon-thyme oil. Serve immediately. Serves 4.