Bring a large pot of heavily salted water to a boil. Cook pasta until al dente according to package directions. Reserve 1/2 cup of the starchy pasta water, then drain the pasta and return it to the pot off the heat.

In a medium bowl, whisk together the room-temperature yogurt, minced garlic, salt, and pepper until smooth. Set aside.

Set a large skillet over medium-high heat. Add the chopped onion and cook, stirring, until softened, about 3 minutes. Add the ground lamb and break it up with a spoon. Cook until deeply browned and internal temperature reaches 160°F (71°C), 6–8 minutes. Stir in the tomato paste, cumin, Aleppo pepper, and salt. Pour in the water, scrape up any browned bits, and let it simmer for 5 minutes until rich and slightly thickened.

In a small saucepan, melt the butter over medium heat. Once it foams, add the smoked paprika and crushed dried mint. Swirl the pan for 30–60 seconds until the mixture is fragrant and the paprika sizzles. Remove from heat immediately to prevent burning.

Working quickly, pour the yogurt sauce over the hot pasta and toss gently to coat, adding a splash of reserved pasta water if needed to loosen it. Divide among bowls. Top each portion generously with the spiced lamb mixture. Drizzle the sizzling paprika-mint butter over everything. Finish with a flurry of fresh mint and chili flakes to taste.