Place a rack in middle of oven; preheat to 475°F. 

Whisk half of the garlic, ½ cup oil, 1 tablespoon of mint, and 1 tablespoon of oregano in a small bowl. 

Brush both sides of eggplant rounds with herb oil, making sure to get all the herbs and garlic onto eggplant, season with salt and pepper. Transfer eggplant to a rimmed baking sheet (it’s okay to pile the rounds on top of each other) and roast until tender and browned, 35–45 minutes. Reduce oven temperature to 400°F.

Meanwhile, heat remaining 2 tablespoons of oil in a large wide pot over high. Cook lamb, breaking up with a spoon, until browned on all sides and cooked through and liquid from meat is evaporated (there will be a lot of rendered fat), 12–16 minutes. Strain fat through a fine-mesh sieve into a clean small bowl and transfer lamb to a medium bowl. Reserve 3 tablespoons of lamb fat; discard remaining fat.

Heat 2 tablespoons of lamb fat in same pot over medium-high (reserve remaining 1 tablespoon of lamb fat for assembling the moussaka). 

Add onion, cinnamon, 2½ teaspoons of salt, and ½ tablespoons of pepper and cook, stirring occasionally, until onion is tender and translucent, 8–10 minutes. 

Add chiles and remaining garlic and cook, scraping up any browned bits from the bottom of the pot, until onion is golden brown, about 5 minutes. 

Add paprika and tomato paste and cook until brick red in color, about 1 minute. 

Add wine and cook, stirring occasionally, until slightly reduced and no longer smells of alcohol, about 3 minutes. 

Add tomatoes and break up with a wooden spoon into small pieces (the seeds will shoot out at you if you’re too aggressive, so start slowly—puncture the tomato, then get your smash and break on!). 

Add lamb and remaining 1 tablespoon of mint and 1 tablespoon of oregano and cook, stirring occasionally, until most of the liquid is evaporated and mixture looks like a thick meat sauce, 5–7 minutes. Pluck out and discard cinnamon stick.

BÉCHAMEL AND ASSEMBLY

Heat butter in a medium saucepan over medium until foaming. 

Add flour and cook, whisking constantly, until combined, about 1 minute. Whisk in warm milk and bring sauce to a boil. Cook béchamel, whisking often, until very thick (it should have the consistency of pudding), about 5 minutes; stir in salt. 

Remove from heat and whisk in farmer cheese and half of the Pecorino. Let sit 10 minutes for cheese to melt, then add egg yolks and vigorously whisk until combined and béchamel is golden yellow.

Brush a 13 x 9 baking pan with remaining 1 tablespoon of lamb fat. Layer half of eggplant in pan, covering the bottom entirely. Spread half of lamb mixture over eggplant in an even layer. Repeat with remaining eggplant and lamb to make another layer of each. Top with béchamel and smooth surface; sprinkle with remaining Pecorino.

Bake moussaka until bubbling vigorously and béchamel is browned in spots, 30–45 minutes. Let cool 30 minutes before serving.

Moussaka can be baked 1 day ahead. Let cool, then cover and chill, or freeze up to 3 months. Thaw before reheating in a 250° oven until warmed through, about 1 hour.