In a medium bowl, whisk together the sesame oil, egg white, soy sauce, and cornstarch until smooth. Add the chicken, turning the pieces to coat them completely in the sauce. Leave to marinate for 15 minutes.

In 4-5 qt Dutch oven or soup pot, pour enough vegetable oil to cover the bottom of the pan in a very shallow pool. Turn heat to medium-high. When the oil is actively shimmering carefully lay the marinated chicken pieces in the oil. Don't crowd the pieces - you might need to cook the chicken in two batches. Let the chicken cook completely through, turning one or two times. Transfer the cooked chicken to a plate lined with paper towels to drain. (Repeat with the second batch of chicken if necessary.) Reduce heat to medium.

If there are chicken bits (fond) stuck to the pan, deglaze the pot before proceeding. Use a splash of the chicken broth or white wine to scrape up the bits. If there's a lot of fond, remove some from the pot and discard (this is up to you).

Add another tablespoon of the vegetable oil and let it heat until shimmering. Add the onion, peppers, broccoli stems (not florets), and saute until all are soft, about 8 minutes. Add the garlic and stir until fragrant (just 30 seconds or so).

Add 1 cup of the chicken broth, the soy sauce, hoisin, ginger, cornstarch, and sambal oelek, and stir well. Let simmer for several minutes, until thick and glossy. Tip in the remaining 3 cups of chicken broth, and bring the soup to a gentle simmer. Stir in the broccoli florets. Adjust the heat so that the soup bubbling very gently, and cook for 10 minutes.

Reduce the heat to low, and dump in the cooked rice, stirring thoroughly to mix. Taste the soup - it should not need salt, but if you want it slightly sweeter, add another teaspoon or two of hoisin (completely optional).

When the chicken is cool enough to handle, slice or snip with kitchen shears into spoon-sized pieces. Add to the soup. Top with the scallions, and serve immediately.