Place a large non-stick skillet or Dutch oven over medium-high heat. Add the lean ground beef and cook until fully browned, breaking it into small crumbles. Drain any excess fluid.
Incorporate the diced onion into the skillet and sauté for 4 minutes until translucent.
Add the minced garlic, tomato paste, mustard, smoked paprika, salt, and black pepper. Stir constantly for 1 minute to caramelize the tomato paste and bloom the spices.
Add the dry orzo to the skillet, stirring for 1 minute to incorporate with the fat and aromatics.
Deglaze the pan with the beef broth and add the milk. Bring the mixture to a light boil.
Reduce heat to medium-low, cover with a tight-fitting lid, and simmer for 9 to 11 minutes. Stir every 3 minutes to prevent the orzo from adhering to the bottom of the pan.
Once the orzo is al dente and the liquid has reduced to a thick sauce consistency, remove the pan from the heat source.
Stir in the shredded cheddar cheese until fully melted and integrated.
Gently fold in the Greek yogurt until the sauce is uniform and creamy. Avoid boiling after adding yogurt to prevent curdling.
Adjust seasoning to taste and serve immediately.