Clean the chicken, place it in a large pot, and add enough water to cover it. Bring to a boil over high heat, skimming off any foam that rises to the surface. Once the water reaches a boil, add the onion, cinnamon stick, cardamom pods, and bay leaves. Reduce the heat to low, cover the pot, and simmer for about 2 hours, or until the chicken is fully cooked and tender enough to fall off the bone.
Strain the broth and transfer the chicken to a dish. Once cool enough to handle, remove the bones and shred the chicken into large pieces.
Break the vermicelli into 2.5 cm (1-inch) pieces, or use a brand that is already cut into smaller pieces.
Heat the oil and butter in a large pot. Add the chopped onions and sauté until softened and tender. Add the vermicelli and stir constantly to prevent it from burning. Continue stirring and sautéing until the vermicelli turns golden brown. This happens faster than you might expect, so keep a close eye on it!
Add the bulgur and stir briefly to coat it with the oil, vermicelli, and onions. Sauté everything together for about 30 seconds.
Pour in 4 cups of the strained chicken broth, season with salt and freshly ground black pepper, and bring to a boil. Add the shredded chicken and give everything a quick stir. Cover the pot, reduce the heat to very low, and let it simmer for about 25 minutes, allowing the flavors to meld beautifully.
Fluff the chicken bulgur pilaf with a fork and serve warm alongside plain yogurt or a refreshing Middle Eastern salad. Heavenly delicious!