Heat the sesame oil and vegetable oil together in a large skillet over medium-high heat. Add the diced chicken along with a pinch of the salt and pepper. Let it cook for four to six minutes, stirring occasionally, until it's fully cooked through and lightly golden on the edges. Move it to a plate and set it aside for now.
Using the same skillet, without wiping it out, add the frozen peas and carrots straight from the freezer. Sauté them for about two minutes, just until they're heated through and slightly tender. I like keeping the pan on medium-high here so the vegetables pick up a little color instead of steaming and turning soft.
Push the vegetables to one side of the pan to clear some open space. Pour the beaten eggs into that empty side and let them sit for a few seconds before stirring gently. Scramble them until just set, then break them into smaller pieces and mix them in with the vegetables already in the pan.
Return the cooked chicken to the skillet along with the rice, soy sauce, remaining salt, and pepper. Stir everything together so the rice breaks up and every piece gets coated in sauce. Let it cook for three to five minutes, stirring occasionally, until the rice is heated through and starting to crisp slightly on the bottom.
Turn off the heat and scatter the sliced green onions over the top. Give it one more quick stir, then serve it straight from the skillet while it's hot. This whole meal comes together in one pan, so cleanup is just as easy as the cooking.