Melt the butter in a lidded skillet over medium-low heat. Cook and stir orzo pasta until golden brown. Stir in onion and cook until onion becomes translucent, then add garlic and cook for 1 minute. Mix in the rice and chicken broth. Increase heat to high and bring to a boil. Reduce heat to medium-low, cover, and simmer until the rice is tender, and the liquid has been absorbed, 20 to 25 minutes. Remove from heat and let stand for 5 minutes, then fluff with a fork.