Heat oven to 400 degrees. Scatter carrots, onions and tomato halves in the bottom of two roasting pans, taking care not to crowd. Toss with 2½ T olive oil and season with salt and pepper.
Slice about one inch off the top of the garlic clove, leaving the peels on. Drizzle a ½ T olive oil over the slice and replace top. Wrap in tin foil and add the garlic head to one of the roasting pans.
Roast the garlic and vegetables for 10 minutes and then turn all the vegetables with a spatula, except the tomatoes. Return all the vegetables to the oven and roast another 10 minutes. Remove the carrots and onions and put into a large heavy-bottomed pot. Return the tomatoes and the garlic to the oven for another 20 minutes.
While the tomatoes and garlic continue to roast add 6 cups of stock, ginger, lime zest and lime juice to the pot with the carrots and the onions. Bring to a boil and reduce heat to a simmer until the tomatoes and garlic are fully roasted. Add the tomatoes and squeeze out the soft roasted garlic into the soup pot, stir in and remove from heat.
Puree the soup using a blender, a food processor or a stick blender. A blender will provide the smoothest results. Return the pureed soup to the pot and stir in the coconut milk. Taste and season with salt and pepper.
Serve with an optional dollop of yogurt and a squeeze of lime.