Very lightly sprinkle salt over the chicken thighs. Let them sit for 10 minutes, then pat them dry with a paper towel to remove any moisture.
Prepare the vegetable filling. Cut the carrot into 1/2-inch (1 cm) thick and 3-inch (7.5 cm) long pieces. Trim the ends of the green beans. Boil water in a pot, add the carrot, and cook for 3 minutes. Then, add the green beans and cook for another 2 minutes. Season with salt, soy sauce, and mirin. Continue cooking until half of the liquid has evaporated. Let the vegetables cool.
Make shallow cuts on the surface of the chicken thighs. Pound them with a meat mallet to thin them out. Place 3 strips of carrot and 3 green beans at one edge of a chicken thigh, then roll it up from one end to the other. Secure with a toothpick. Repeat with the remaining chicken thigh.
Heat oil in a frying pan over medium heat. Place the chicken rolls in the pan and cover with a lid. Cook, turning the chicken every couple of minutes, until evenly browned on all sides. Add soy sauce, sake, and mirin, and cook for a few more minutes until the sauce slightly thickens. Slice the chicken rolls and serve.