In a large pot over medium heat, heat oil. Add onion, carrots, and celery and cook, stirring, until softened, about 7 minutes. Add garlic and cook, stirring, until fragrant, about 1 minute more. Add broth, oregano and thyme; season with salt and pepper. Bring to a boil. Add rice and reduce heat to medium-low. Simmer, stirring occasionally, until rice is cooked through, about 20 minutes. Add chicken and return to a simmer.
In a medium bowl, whisk egg yolks and lemon juice. Slowly add ¼ c. broth from pot to egg mixture while whisking. Slowly whisk egg mixture back into the pot. Let simmer, stirring occasionally, until thickened, about 5 minutes.
Divide soup among bowls. Top with dill and crushed red pepper flakes. Serve with lemon wedges alongside.
