Heat a heavy-bottomed pan over medium heat and add the butter along with the olive oil. Once the mixture is hot, incorporate the finely chopped onion and a small pinch of salt, allowing it to cook gently for 4 to 5 minutes.

Introduce the garlic and carrots, stirring occasionally, and continue to cook until the garlic softens and its aroma becomes pronounced.

Stir in the dry orzo and pour in the broth. Bring the mixture to a simmer, then cover with a lid and let cook undisturbed for approximately 6 minutes.

Remove the lid, add the chopped broccoli and milk, stir to combine, and cover again. Cook for an additional 4 minutes or until the orzo reaches an al dente texture and the broccoli is tender.

Turn off the heat, stir in the cheddar and Parmesan until melted and creamy. Season with salt and freshly ground black pepper according to preference. Serve immediately.