Combine 1/2 cup of the broth, the water, and the salt in a small, heavy saucepan. Bring to a boil and stir in the rice. Reduce the heat to low, cover the pan, and simmer until all the liquid is absorbed and the rice is just tender, 20 to 22 minutes; set aside.
In a large, heavy saucepan, bring the remaining 5-1/2 cups broth to a simmer over medium heat. Add the chicken, reduce the heat to a gentle simmer, and poach until the chicken is firm and just cooked through, 3 to 5 minutes. (If the broth is cloudy, you can remove the chicken, strain the liquid through a cheesecloth-lined sieve, and return the chicken to the broth.)
Add the reserved rice and the lemon juice and season with salt and pepper. Sprinkle on the mint and serve right away.