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 put it 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, 8 to 10 minutes, deglazing the pot with a splash of broth if a lot of food sticks.
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.
Add the rinsed rice and toss to coat for 1 minute, deglazing with a splash of broth as needed. Add one can of drained chickpeas, plus ¼ teaspoon kosher salt and pepper to taste. Toss to coat. Pour in the broth, followed by half of the chopped dill. Bring to a boil.
Cover the pan, and reduce the heat to low to maintain a simmer. Simmer for 15 minutes, stirring occasionally, 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 as needed.
Stir the Chickpea Puree into the soup, then add the chopped spinach. Simmer for 1 to 2 minutes, stirring frequently, 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.