Preheat your oven to 400°F. Arrange sweet potato wedges on a large baking sheet in a single layer without overcrowding. Drizzle generously with olive oil and season with salt and pepper. Roast for 30 minutes, flipping halfway through at the 15-minute mark, until edges are deeply caramelized and golden brown with dark spots. Set aside a few pieces for garnish if desired.

Heat 2 tablespoons of olive oil in a large pot (at least 4-quart capacity) over medium heat. Add diced onion and cook for about 5 minutes, stirring occasionally, until softened and translucent but not browned. Add minced garlic, dried thyme, rosemary, and basil. Cook for 1 to 2 minutes until fragrant, being careful not to burn the garlic.

Add all roasted sweet potatoes to the pot (except any reserved for garnish), followed by the vegetable broth. Stir to combine and bring to a boil over high heat. Reduce heat to medium-low and simmer gently for 10 to 15 minutes to allow flavors to meld and any firm pieces to soften completely.

Using an immersion blender, blend the soup directly in the pot until completely smooth and creamy, about 2 to 3 minutes of continuous blending. Alternatively, transfer in batches to a regular blender, filling only halfway and holding the lid down firmly with a kitchen towel to prevent steam pressure buildup.

Stir in the coconut milk or heavy cream and heat through gently for about 5 minutes over low heat, stirring occasionally. Avoid boiling to prevent curdling or separation. Taste and adjust seasoning with salt and pepper as needed, typically ½ to ¾ teaspoon of salt depending on broth.

Ladle soup into bowls and garnish with reserved roasted sweet potato wedges and a fresh sprig of thyme. Serve warm.