Heat the olive oil and butter in a large pot or deep skillet over medium-high heat until the butter melts.
Add the chopped onion and sauté for 3–4 minutes, until softened.
Stir in the minced garlic and the uncooked orzo and cook for 2–3 minutes, stirring frequently to lightly toast the orzo.
Pour in the chicken or vegetable broth and bring to a gentle boil, then reduce heat to maintain a gentle simmer. Cook uncovered for about 10 minutes, stirring often, until the orzo is mostly cooked and the liquid is mostly absorbed; reduce heat if it is boiling too vigorously and add a splash more broth if needed.
Stir the softened Boursin cheese into the pot until the sauce is smooth and evenly combined.
Remove the pot from the heat, then stir in the baby spinach and chopped or shredded cooked chicken. Cover and let sit for 5 minutes so the spinach wilts and the chicken heats through.
Season to taste with salt and pepper, stir gently, and serve topped with chopped fresh chives if desired.