Position a rack in the centre of the oven and preheat the oven to its highest temperature.
Pour 2 tablespoons of the oil into a heavy ovenproof pan big enough to hold the carrots in a single layer. Set the pan over high heat and bring the oil to a light smoke. Add the carrots, sprinkle over a teaspoon of sea salt and turn the carrots to coat them in the oil.
Cook, turning over the carrots occasionally, until they’re browned in spots, this should take 6-8 minutes. Put the pan in the oven and roast, shaking the pan occasionally, until the carrots are evenly tender, this should take 8-12 minutes, depending on the size of the carrots. Remove from the oven and let the carrots cool slightly.
Pluck enough 5-8cm/2-3in delicate sprigs from the reserved carrot tops to make 40g/1½oz and toss them in a bowl with the basil leaves.
Whisk together the remaining 2 tablespoons of oil with the lemon juice and a good pinch of sea salt in a small bowl until the mixture looks creamy. Use a little of this lemon dressing to lightly dress the carrot-top and basil mixture (set aside the remaining lemon dressing). Sprinkle with a little more sea salt and toss well.
To make the pesto, put the carrot tops and basil in a small food processor and pulse several times. Add the walnuts, Parmesan, garlic and sea salt. Pulse several more times, add some oil, then blend, stopping and scraping down the sides of the processor if needed. Add enough oil to loosen the mixture and blend until well combined but still a bit chunky. Taste and season with more sea salt, if needed.
To serve, put the Burrata slices on a platter, arrange the roasted carrots on the platter and dollop over the pesto. Arrange the carrot-top and basil mixture on top and drizzle everything with the remaining lemon dressing.