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1
Position 1 rack in center and 1 rack in bottom third of oven; preheat to 500u00b0F.
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2
Line 2 large baking sheets with foil.
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3
Cut onions crosswise nearly, but not quite, all the way through. Make two or three more sectional cuts - always stopping before you cut through, so that onion is in 3/4-inch-thick wedges (the onion stays attached at one end and resembles an onion 'flower' (with very thick petals ;-) ).
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4
Place in medium bowl; toss with oil.
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5
Arrange onions, cut side down, on baking sheets.
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6
Sprinkle with salt and pepper.
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7
Roast until onions are brown and tender, rotating pans in oven and turning onions once, about 45 minutes (they will caramelize and be quite dark in parts but, if you have coated them well with olive oil, will not have a 'burnt' taste).
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8
Meanwhile, melt butter in heavy small saucepan over medium-high heat.
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9
Add sugar and stir until sugar dissolves.
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10
Remove from heat.
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11
Add vinegar.
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12
Return to heat.
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13
Simmer until mixture thickens slightly, about 2 minutes.
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14
(Onions and balsamic glaze can be made 1 day ahead. Cool. Cover separately and chill. Rewarm onions in 375u00b0F oven about 15 minutes. Stir glaze over low heat to rewarm.) Arrange onions on platter.
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15
Drizzle glaze over.
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16
Sprinkle with parsley.
