Heat the oil and butter in a large pot or deep skillet over medium-high heat until the butter is melted.
Add the chopped onion and sauté for 3–4 minutes, until softened.
Stir in the minced garlic and uncooked orzo and cook for 2–3 minutes, stirring frequently, until the orzo is slightly toasted and fragrant.
Pour in the chicken or vegetable broth and bring to a gentle boil, then reduce heat to medium or medium-low so the liquid simmers gently.
Cook uncovered for about 10 minutes, stirring often, until most of the liquid is absorbed and the orzo is al dente; add a splash more broth if it is reducing too quickly.
Stir the softened Boursin cheese into the pot until it melts and creates a creamy sauce.
Remove the pot from the heat, add the spinach and cooked chicken, then cover and let sit for 5 minutes to wilt the spinach and warm the chicken.
Uncover, season with salt and pepper to taste, and garnish with chopped fresh chives if desired before serving.
