Place the rice and 3 cups of water in a large pot and bring to simmer. Cover and cook for about 40 minutes until the rice is fully cooked and has fallen apart. If after 40 minutes, the rice hasn't fallen apart, add one more cup of water and cook for 15 more minutes.
In a large bowl, mix yogurt, egg, flour and salt.
Once the rice is cooked and has broken into very small pieces, turn the heat off.
using ladle, slowly add about 1 ½ cup of the rice and water to the yogurt and egg mixture and stir well after each addition.
Add the yogurt and egg mixture back into the pot. Stir well, turn the heat on to medium and bring to a low simmer.
Simmer for 15 minutes, stirring occasionally.
In a small pan, melt the butter and saute dried mint for about 30 seconds.
Add sauteed mint and butter to the simmering soup. Stir well.
Serve Turkish yogurt soup warm.