Heat oven to 375 degrees

Heat 3 tbsp. butter and the oil in a 6-qt. saucepan over medium-high heat.

Season lamb with salt and pepper and toss with 1⁄4 cup flour.

Cook lamb, turning as needed, until browned, 10–12 minutes.

Add rice, garlic, and 1⁄2 cup water; boil.

Reduce heat to medium-low; cook, covered, until rice is just tender, about 15-20 minutes.

Stir in oregano, salt, and pepper and transfer to a 9" x 13" baking dish; set aside.

Melt remaining butter in a saucepan over medium-high heat.

Whisk in remaining flour; cook until smooth, 2 minutes.

Remove from heat; whisk in yogurt, nutmeg, eggs, salt, and pepper until smooth.

Pour yogurt sauce evenly over lamb mixture.

Bake until golden and the lamb is tender, 45–60 minutes.