Heat 2 tablespoons oil in a large skillet over high heat.
Add onions and 1/4 teaspoon salt; cook, stirring frequently, until the onions begin to brown, about 5 minutes. Reduce the heat to low, add 2 tablespoons water and cover. Cook, stirring frequently until the pan cools down, and then occasionally, always covering the pan again, until the onions are greatly reduced and have a deep caramel color, 25 to 30 minutes.
Meanwhile, combine the remaining 3 cups water and 3/4 teaspoon salt in a soup pot or Dutch oven; add rice. Bring to a boil. Reduce heat to maintain a simmer, cover and cook for 15 minutes. Trim the white ribs out of the chard (save for another use, such as to add to a stir-fry or other soup). Coarsely chop the chard greens and spinach.
When the rice has cooked for 15 minutes, stir in the chard greens. Return to a simmer; cover and cook for 10 minutes. When the onions are caramelized, stir a little of the simmering liquid into them; add them to the rice along with the spinach, broth and cayenne. Return to a simmer, cover and cook, stirring once, until the spinach is tender but still bright green, about 5 minutes more.
Puree the soup in the pot with an immersion blender until perfectly smooth or in a regular blender in batches (return it to the pot). Stir in 1 tablespoon lemon juice. Taste and add more lemon juice, if desired. Garnish each bowl of soup with a drizzle of olive oil.