Heat the olive oil in a large pot or deep skillet over medium-high heat.
Add the ground beef and cook, breaking it up with a spoon, until browned and cooked through, about 5-7 minutes.
Add the chopped onion and cook until softened, about 3 minutes.
Stir in the minced garlic, dried oregano, dried thyme, salt, and black pepper. Cook for 1 minute until fragrant.
Add the orzo pasta to the pot and stir to coat it with the beef mixture.
Pour in the beef broth and bring the mixture to a boil.
Reduce heat to medium-low, cover, and simmer for 10-12 minutes, stirring occasionally, until the orzo is tender and most of the liquid is absorbed.
Stir in the milk, grated Parmesan cheese, and butter until the mixture is creamy and well combined.
Cook for an additional 2-3 minutes, stirring frequently, until heated through and creamy.
Remove from heat and stir in the chopped fresh parsley.
Serve immediately.