Heat the oil in a small (3-qt.) Dutch oven or soup kettle over medium heat. Add the onion and cook until softened, 5 to 7 minutes. Add the chicken thighs, broth, rice, and oregano. Cover, raise the heat to medium high, and bring to a full boil; turn off the heat and let stand, covered, for 5 minutes. With tongs, transfer the chicken to a cutting board and shred it with a table knife and fork, discarding any obvious fat or gristle. Working in two batches, purée the broth, rice, and onions in a blender until very smooth, 30 to 60 seconds.
Return the chicken and the thickened broth to the pot and bring to a simmer over medium-low heat. Add the spinach and dill; cook until the spinach wilts completely, about 3 minutes. Stir in the lemon juice. Season to taste with salt, pepper, and more lemon juice.