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Put the chicken in a large bowl with 2 tablespoons oil, mirin, corn starch, 1 teaspoon kosher salt, and egg white. Mix thoroughly and leave to marinate for 10 minutes.
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Heat ¼ cup oil in a wok or large skillet over medium-high heat. Add chicken and cook, stirring often, until chicken is almost cooked through, about 5 minutes. Remove chicken to a plate with a slotted spoon and pour out any excess coating from the wok.
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Add remaining 1 tablespoon oil and turn heat down to low. Add garlic, ginger, and chilies, and cook, stirring often, until fragrant, about 2 minutes. Add bell pepper, cucumber, and carrot. Turn heat back to medium-high. Cook, stirring often, for 3 minutes.
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To the vegetables, add soy sauce, oyster sauce, remaining 1 teaspoon kosher salt, chicken bouillon in water, sugar, and ground white pepper. Stir well. Return the chicken to the wok and continue cooking, stirring often, until vegetables are crisp tender and chicken is cooked through, about 3 minutes.
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Add scallions, peanuts, and sesame oil. Mix well and cook 1 more minute. Serve with additional scallions or peanuts for garnish.