Heat oil in a skillet over medium heat. Add chicken breast; cook just until no longer pink in the center; about 3 minutes. Transfer chicken to a plate.
Whisk coconut milk and curry paste together in skillet until combined. Add potato; simmer until potatoes begin to soften, about 10 minutes. Stir cooked chicken, peas and carrots, fish sauce, sugar, and lime leaves into the curry mixture; simmer until potatoes are soft, about 10 minutes more.