Put the cup of rice in the pan with 2 cups of water in medium heat ( 2 cups of water for 1 of rice is the perfect measure).
Mix the 2 table spoons of Greek yogurt, the egg yuk and the lemon juice until you have an homogenous mix.
Add the previous mix to the rice (which is almost cooked) , you can add water of the rice with a spoon to get all the mix in the cup. Then just stir smoothly in the pan and low the heat.
Aside, put in low heat some butter in another pan. Once butter is melted add the mint, and mix them together. This will not take more than a minute, so be careful not to burn the butter.
When butter is mixed with mint, add immediately to the rice with the yogurt mix added previously, and stir smoothly. Then just let it cold a little, and you're done!