Rinse the rice in cold water until the water runs clear. Soak the rice in the cold water for 30 minutes, then drain for 15 minutes.
Cook the rice in a rice cooker according to manufacturer's instructions or place the rice and 1 1/4 cups of water in a heavy bottomed saucepan. Bring to a boil over medium-high heat, then reduce the heat to the lowest setting, cover, and cook until all the water has been absorbed, 10-13 minutes. Remove from heat, leave the cover on, and allow to rest for another 10 minutes.
Remove the cover, fluff the rice with a fork or rice scooper and allow to rest until just cool enough to handle. It should still be warm to hot. Pack the freshly steamed, warm rice in the mold (making sure not to pack the rice too densely), put the two sides together and squeeze using the handle to form the cat's head. If forming by hand, wet your hands with cold water, rub with salt and form the mound of rice into a cat's head, using plastic wrap if needed. Use the nori punch to cut out the desired cat face or cut out with scissors by hand.
Line the bottom tier of the bento box with lettuce. Add the cat (nekonigiri. Fill the second tier with Mochiko Fried chicken, grape tomatoes, shelled edamame, and carrots. Cut fish out of the cheese slice using a cutter or by hand with a knife.
Heat the instant miso packet according to package instructions right before serving.