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1. Prep your veggies. First peel and dice carrots and sweet potato, and set aside. Dice pepper and add to bowl with veggies. Dice the onions and then saute them in a medium sauce-pan for about five minutes, until they begin to look transparent.
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2. Once the onions are ready, you can add your vegetables. Cook, stirring frequently so that they do not stick to the bottom of the pan, for five to seven minutes.
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3. Add water and bring to a boil. Once soup has come to a boil, lower the heat to a simmer and cover the pot to let the vegetables cook for ten to fifteen minutes.
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4. While waiting for the vegetables to soften, prepare the pesto mixture. In a small bowl, mix olive oil, basil, and oregano. Add the egg yolk and whisk. Lastly, whisk in the tomato paste and mix until thoroughly combined.
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5. Once the vegetables are soft (you should be able to mash or easily poke them with a fork) it is time to combine the broth with the pesto. This must be done gradually to prevent the egg yolk from curdling. With a ladle, spoon small amounts of hot broth into the bowl of pesto, whisking to combine. Continue adding until the bowl of pesto is hot (it should be steaming), at which point you can add the pesto to the pot of soup.
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6. Add salt and pepper (however much you like) and puree the soup using an immersion blender (or whatever works best for you) and blend until the soup has a thick consistency and there are no visible chunks of vegetable.
