In a large dutch oven or oven-proof casserole, heat the oil over medium heat. Add the carrots, zucchini, and scallions and cover. Cook, stirring often, until the vegetables soften, about 3 minutes. Add the garlic and cook 1 minute.

Add the chicken broth and bring to a boil over high heat. Season the chicken breasts with 1/4 teaspoon salt and 1/8 teaspoon pepper. Add the chicken breasts and return the liquid to a boil. Reduce the heat to low and cover. Simmer until the chicken is firm and cooked through, about 12 minutes. Transfer the chicken breasts to a cutting board and cover with aluminum foil to keep warm.

In a small bowl, whisk the egg, lemon juice, cornstarch, and herbs to dissolve the cornstarch. Gradually stir about 1/2 cup of the hot cooking liquid into the egg mixture. Stirring constantly, pour the egg mixture into the cooking liquid. Cook just until the sauce is lightly thickened but not simmering, about 1 minute. Remove from the heat. Season with salt and pepper to taste.

Spoon the rice onto individual dinner plates and top with the sauce. Using a sharp knife, cut each breast diagonally into 4 or 5 thick slices, slip the knife under the sliced breast, and transfer to the bowl of rice, fanning out each breast slightly. Serve immediately.