Add the rice to a fine-mesh sieve and rinse a few times until almost clear, then drain.
Drain and rinse one can of chickpeas and set aside for step 5. Drain and rinse the other can of chickpeas and transfer to a food processor or small-capacity blender for step 8.
In a large saucepan or Dutch oven, heat the olive oil over medium-high heat. Once shimmering, add the carrots and onion. Season with a few pinches of salt. Stir occasionally, until softened and lightly browned, about 10 minutes.
Add the garlic and cook for 2 minutes, stirring frequently. Add the tomato paste plus spices (oregano, coriander, turmeric, and red pepper flakes). Stir almost constantly for 1 to 2 minutes until the tomato paste is a darker red brick color. If it sticks or starts to burn, lower the heat and deglaze with a splash of broth.
Add the rinsed rice and toss to coat in the aromatics and oil for 1 minute. Add one can of drained chickpeas, plus ¼ teaspoon kosher salt and pepper to taste. Toss to coat. Pour in the vegetable broth, followed by half of the chopped dill. Bring to a boil.
Reduce the heat to low to maintain a simmer and cover the pan. Simmer, covered, for 15 minutes, stirring once or twice, or until the rice is soft.
While the soup simmers, roughly chop the spinach and make the chickpea puree.
Chickpea Puree: To the chickpeas in the food processor or blender, add the milk, lemon zest, 2 tablespoons lemon juice, nutritional yeast, miso, 1 teaspoon kosher salt, and pepper to taste. Blend until pureed and creamy, scraping down the sides as needed. Taste for seasonings, adding more salt or lemon juice as needed.
Stir the Chickpea Puree into the soup to combine, then add the chopped spinach. Simmer for 1 to 2 minutes, or until the soup thickens and greens slightly wilt.
Off the heat, stir in the remaining dill and 1 tablespoon of lemon juice. Taste, adding salt/pepper or more lemon juice as desired. Transfer to bowls and drizzle each bowl with a bit of good-quality olive oil for richness.