Preheat your oven to 400°F (200°C). Line a baking sheet with parchment paper.
Wash and peel the carrots, then cut them into approximately 1-inch by 1/2-inch pieces.
Toss the carrots with the olive oil until evenly coated. Spread them in a single layer on the prepared baking sheet. Roast for 15 minutes.
Remove the baking sheet from the oven and sprinkle 2 teaspoons of fresh dill over the carrots. Return to the oven and roast for another 15 minutes, or until the carrots are very soft and easily mashed with a fork.
Allow the carrots to cool slightly. Transfer them to a blender or food processor with the remaining 2 teaspoons of dill, coconut milk, and water. Blend until smooth, adding a little more liquid as needed to reach your desired consistency.
Allow the puree to cool to a safe serving temperature. Check the texture before serving to ensure it is appropriate for your baby's developmental stage.