In a large pot or Dutch oven, heat the oil over medium-high heat. Add the leek and carrot and cook, stirring frequently, until just beginning to soften, 3 to 4 minutes.
Add the garlic and cook, stirring constantly, until fragrant, 30 seconds. Add the sweet potato, 1 cup of water, the broth, bay leaves, salt, and pepper, and bring to a simmer.
Simmer, lowering the heat if you need to, until the sweet potatoes are tender, about 10 minutes.
Add the broccoli, cauliflower, and kale and cook until the broccoli is bright green and tender but not soggy, 2 to 3 minutes.
Stir in the parsley and thyme. Taste the broth and season with more salt and pepper if needed