Whisk together mayonnaise, tomato paste, melted butter, garlic powder, paprika, sugar, and water in a small bowl until completely smooth. Cover and refrigerate.
Heat 1 tablespoon of vegetable oil in a large wok or skillet over high heat. Add the sliced beef and diced chicken, cooking until browned and cooked through (about 5-7 minutes). Remove proteins from the pan and set aside.
In the same wok, add the remaining 1 tablespoon of oil. Sauté the diced red bell pepper and peas for 2-3 minutes until tender.
Push the vegetables to one side of the pan. Pour the beaten eggs onto the empty side and scramble until fully cooked.
Add the day-old rice, cooked beef, and chicken back into the wok. Pour in the soy sauce and stir-fry everything together for 3-4 minutes until heated through and well combined.
Serve the stir-fried rice hot, drizzled generously with the chilled creamy dressing.