Preheat the oven to 375˚. Melt 2 tablespoons butter in a medium ovenproof nonstick skillet over medium-high heat. Add the leeks and a large pinch of salt and cook, stirring occasionally, until tender but not browned, 5 to 7 minutes; remove to a plate.
Add 1 tablespoon butter and the mushrooms to the skillet. Cook, stirring occasionally, until tender and browned around the edges, 5 to 7 minutes. Add the thyme, garlic and a large pinch of salt. Cook, stirring, until the garlic is tender but not browned, about 30 seconds. Transfer to the plate with the leeks and wipe out the skillet; set aside.
Whisk the eggs with 1/3 cup half-and-half in a large bowl. Add the leek-mushroom mixture, chives, 1 teaspoon salt and a few grinds of pepper. Mix the goat cheese and remaining 2 tablespoons half-and-half in a small bowl until smooth and creamy.
Melt the remaining 1 tablespoon butter in the reserved skillet over medium heat. Add the egg mixture and arrange the meatballs in the skillet, pressing them about halfway into the egg mixture. Drop small spoonfuls of the goat cheese mixture evenly over the eggs. Transfer the skillet to the oven and bake until the frittata is lightly browned, set and puffed in the center, 25 to 30 minutes. Slide onto a cutting board and let rest 10 minutes before slicing.