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Cook the rice: Add the rinsed rice and 1 cup water to a pot. Cook until the rice is tender and the water is mostly absorbed.
Build the broth base: Add the chicken broth and 2 tablespoons olive oil to the pot with the rice. Bring to a gentle simmer.
Prepare the yogurt base: In a metal or glass bowl, whisk together the yogurt, egg yolks, flour, and salt until completely smooth.
Temper the yogurt: Slowly add 1 cup of hot broth from the pot into the yogurt mixture, whisking constantly so it does not curdle.
Combine: Gradually pour the warmed yogurt mixture back into the main pot, stirring constantly.
Simmer: Add the dried mint. Simmer over low heat for about 10 minutes, stirring occasionally so the rice does not stick to the bottom.
Rest: Turn off the heat and let the soup rest for 10 minutes.
Make the sauce: Heat 2 tablespoons olive oil and red pepper flakes in a small pan just until fragrant (about 30 seconds).
Serve warm or cold in a bowl, drizzling the sauce over the top. Bon appétit.