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1
Scatter eggplant on rimmed baking sheet. Sprinkle with coarse salt; let stand 1 hour, tossing occasionally. Rinse eggplant and pat very dry.
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2
Heat 3 tablespoons oil in heavy large pot over medium-high heat. Add eggplant and saute until tender, about 12 minutes. Transfer to medium bowl.
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3
Sprinkle lamb generously with coarse salt and pepper, then dust with flour to coat. Heat 2 tablespoons oil in same pot over medium-high heat. Add half of lamb. Saute until browned, about 8 minutes. Transfer lamb to large bowl. Repeat with 2 tablespoons oil and remaining lamb.
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4
Add 1 additional tablespoon oil to same pot, if needed. Add onions. Cover and cook over medium-low heat until very tender, about 10 minutes (bottom of pot will be very dark). Add wine to pot. Increase heat and boil until wine evaporates, scraping up browned bits, about 5 minutes. Add tomatoes with juice, broth, garlic, and oregano and bring to boil. Add lamb with any accumulated juices. Cover; reduce heat to low and simmer 1 hour. Uncover and continue to simmer until lamb is very tender and gravy thickens slightly, about 45 minutes. Stir in eggplant. Season with salt and pepper. Transfer to 13x9x2-inch glass baking dish. DO AHEAD:
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5
Cool slightly. Cover and chill.
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6
Preheat oven to 375u00b0F. Cook potatoes in large pot of boiling salted water until tender, about 14 minutes.
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7
Meanwhile, melt butter with oil in medium saucepan over medium-high heat. Add garlic. Saute until fragrant, about 1 minute. Add milk and bring to simmer.
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8
Drain potatoes. Return to pot. Stir over medium heat until excess moisture evaporates. Add milk mixture and mash potatoes until just smooth. Stir in cheese. Season with coarse salt and pepper. Drop potatoes over filling by heaping tablespoonfuls, covering completely.
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9
Bake pie until filling is heated through and topping is golden, about 45 minutes.
