Preheat oven to 425 degrees. Place a rimmed baking sheet in the oven to get hot.
In a medium bowl, mix together the Mirin and lemon juice. Set aside.
Cut the chicken breasts into 1-inch strips and transfer to a large bowl.
Season generously with salt and pepper. Add the flour and toss to coat the chicken pieces.
Remove the baking sheet from the oven. Pour both oils onto the baking sheet and arrange the chicken pieces in the oil. Bake for 10 minutes.
While the chicken is baking, prepare the rice according to package instructions.
Also while the chicken is baking, bring 1 C of water to a boil in a steamer pan. Add the carrots, then the broccolette, and steam for 4 - 5 minutes or until the chicken comes out of the oven.
Remove the chicken from the oven and carefully tip any pan juices from the sheet pan into the bowl with the Mirin and lemon juice. Mix well.
Turn the chicken pieces over and add the steamed carrots and broccolette to the sheet pan with the chicken.
Pour the Mirin mixture over the chicken and vegetables. Sprinkle with the sesame seeds. Return to the oven and bake for 14 minutes.
Transfer the chicken, veggies and all of the pan juices to a large bowl. Give it a quick stir.
Serve the sesame chicken and veggies spooned over rice.