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1
Pat one third of lamb dry and season with salt and pepper.
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2
Heat 1/2 tablespoon butter in a 4-quart heavy pot over moderately high heat until foam subsides, then brown seasoned lamb, turning occasionally, about 6 minutes.
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3
Transfer to a plate and brown remaining lamb in same manner in 2 batches, adding more butter if necessary.
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4
When third batch of lamb is browned, return rest of lamb with any juices to pot and add tomato, water, and salt.
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5
Cook at a bare simmer, covered, stirring occasionally, until meat is very tender, 2 1/2 to 3 hours.
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6
Season with salt and pepper.
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7
Prepare grill for cooking.
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8
Fill a large bowl with cold water and add lemon juice.
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9
Prick eggplant in several places with a skewer or toothpick (to prevent bursting).
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10
Grill eggplant, turning occasionally, until charred all over and tender inside, about 15 minutes.
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11
Remove from heat and cool slightly (eggplant will collapse), then peel with a sharp small knife while still warm, leaving stem intact.
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12
(This makes the eggplant easier to handleit may fall apart otherwise.)
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13
Soak eggplant in lemon water 15 minutes, then transfer to a colander.
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14
Discard stem and drain eggplant flesh 45 minutes.
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15
Melt butter in a 9- to 10-inch heavy skillet over moderate heat, then whisk in flour and cook, stirring constantly, until roux just begins to brown, 3 to 4 minutes.
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16
Add milk in a stream, whisking, and bring to a boil, whisking.
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17
Reduce heat and simmer, whisking occasionally, until thick, about 3 minutes.
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18
Add drained eggplant and stir, mashing with a fork, until incorporated.
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19
Stir in cheese and salt and season with pepper.
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20
Serve eggplant topped with braised lamb.
