Melt butter in a skillet over medium-low heat; cook and stir orzo in the melted butter until golden brown, 3 to 5 minutes. Stir onion into orzo and cook until translucent, 5 to 10 minutes. Cook and stir garlic into orzo-onion mixture until fragrant, about 1 minute.
Mix rice and chicken broth into orzo-onion mixture; bring to a boil. Reduce heat to medium-low, cover skillet, and simmer until rice is tender and liquid is absorbed, 20 to 25 minutes. Remove from heat and stir in spinach and feta. Cover and let stand until spinach is wilted and feta is melted, about 5 minutes. Fluff with a fork.