If you are using fresh peas, clean them and place them in a pot of boiling water. Cook until they turn a vibrant green (about 2 to 3 minutes). Drain and rinse under cold water. If you are using frozen peas, reheat them according to package instructions.
Add 1 cup of the cooked peas to a large measuring cup or tall container, together with 1 tablespoon of extra virgin olive oil, 2 tablespoons of vegan parmesan, 2 teaspoons of lemon zest, 3 small mint leaves, 1/4 cup vegan yogurt, and 1/2 cup water. Using an immersion blender, blend until roughly smooth. Set this aside.
On one burner, place a small pot containing all the vegetable broth over low heat. On a second burner, add 1 tablespoon of extra virgin olive oil and butter to a large skillet over medium-high heat. When it's pretty much fully melted, add shallots, garlic, and salt. Sauté until the shallots are transparent (around 3 minutes). Add rice and stir well until the grains are evenly coated with the oil and butter. Continue stirring regularly for about 3 minutes, until the rice grains begin to toast and turn slightly brown. Deglaze the pan with the wine, stirring the rice and veggies so they are evenly coated. Cook until the wine has cooked off (around 1 minute). Reduce heat to low.
Add two ladlefuls of the hot broth, stirring it into the rice. Continue stirring not continuously but regularly until most of the broth has evaporated (around 5 minutes). Then, pour in 1/3 of the pea purée and incorporate into the rice, together with 1 ladle of broth. Cook, while stirring regularly, until the liquid has nearly evaporated (around 5 minutes). Repeat. You should have 1/3 of the pea purée left. Set this aside (you're saving this for the end). Continue ladling hot broth into the rice, stirring regularly to release the rice starch, until all the broth is used up and your rice grains are tender on the outside, but retaining bite at their center (al dente). Turn off heat.
Add the remaining peas, pea purée, and 2 more tablespoons of grated vegan parmesan cheese. Incorporate into rice and continue stirring for about 30 seconds before serving. Garnish with a drizzle of olive oil, more cheese, fresh cracked pepper, lemon zest, and tiny mint leaves.