Set a medium Dutch oven over medium heat. Add the oil and butter; once the butter foams, stir in the onion, carrot, and celery with a pinch of salt. Cook, stirring occasionally, until the vegetables are glossy and beginning to soften, 6 to 7 minutes. Add the green beans and garlic and cook 1 minute more.
Sprinkle the flour over the vegetables and stir constantly for 2 minutes to form a light blond roux.
Gradually whisk in about 2 cups of the broth until smooth, then add the remaining broth along with the bay leaf, Italian seasoning, black pepper, and salt. Bring to a gentle simmer, partially cover, and cook 12 to 14 minutes, stirring now and then, until the vegetables are tender.
Temper the evaporated milk by whisking in a ladle of hot soup, then stir the warmed milk into the pot. Keep at a bare simmer (do not boil) for 3 to 4 minutes to thicken and turn silky.
Fold in the chicken, rice, and peas. Simmer 3 minutes more until everything is hot. Remove the bay leaf, taste, and adjust seasoning. If the soup gets too thick, loosen with a splash of hot water or broth.
Take the pot off the heat and let the soup rest for 5 minutes before serving so the rice absorbs just enough liquid without turning mushy.