Position a rack in the center of the oven and preheat to 500 degrees F.
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 on 1 teaspoon of the salt, and turn the carrots to coat them in the oil. Cook, turning over the carrots occasionally, until they're browned in spots, 6 to 8 minutes. Pop the pan in the oven and roast, shaking the pan occasionally, until the carrots are evenly tender, 8 to 12 minutes, depending on the size of your carrots. Let the carrots cool slightly. Halve the burrata and arrange the halves on a platter. Arrange the carrots on the platter so they're pointing this way and that. Add the pesto here and there in little dollops.
Pluck enough 2-to 3-inch delicate sprigs from the reserved carrot tops to make about 1 1/2 lightly packed cups and toss them in a bowl with the basil. Whisk together the remaining 2 tablespoons of the oil with the lemon juice and a good pinch of salt in a small bowl until the mixture looks creamy. Use a little of the lemon dressing to lightly dress the carrot top-basil mixture, sprinkle on a little more salt, and toss well. Arrange the mixture on top of the carrots and burrata. Drizzle everything with the remaining lemon dressing and serve.
Combine the carrot tops and basil in a small food processor, pulse several times, then add the walnuts, Parmesan, garlic, and salt. Pulse several more times, add the oil, then process full-on, stopping and scraping down the sides of the processor or stirring gently if need be, until the mixture is well combined but still a bit chunky. Taste and season with more salt, if you fancy.
I treat carrot tops as I would a tender herb, adding little sprigs to salads as I might parsley or dill. And because each bunch of carrots can bring twice the volume in tops, I make pesto. As much as I like the particular flavor of the tops themselves, I also like how they carry the flavor of basil, which comes through quite a bit considering how few leaves you use.
From A Girl and Her Greens by April Bloomfield with JJ Goode. Copyright 2015 April Bloomfield. Excerpted by permission of Ecco, an imprint of HarperCollins Publishers.