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1
Prepare grill
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2
On a rack set about 5 inches over glowing coals toast tortillas lightly a few at a time, turning them once, until they begin to puff on each side but are still soft.
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3
Alternatively, tortillas may be toasted on a stove-top griddle over moderately high heat.
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4
In a large non-stick skillet cook onions in oil over moderately low heat, stirring occasionally, until soft, about 15 minutes.
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5
Transfer onions to a bowl and cool.
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6
Stir in parsley, Parmesan, 1/2 cup mayonnaise, and salt and pepper to taste.
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7
Spread onion mixture on 11 tortillas, stacking them, and top the stack with remaining tortilla.
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8
Wrap cake well in plastic wrap and chill at least 1 hour and up to 4 hours.
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9
Discard plastic wrap and with large sharp knife cut cake into 3/4-inch-wide slices (not wedges).
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10
Turn each slice onto its side and secure ends and middle with wooden picks.
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11
In a small bowl stir together vinegar and remaining 1/4 cup mayonnaise and brush some of mixture on a cut side of each slice.
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12
Grill slices, mayonnaise sides down, until golden brown, 3 to 4 minutes.
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13
Brush tops with remaining mayonnaise mixture, turn slices, and grill until golden brown, 3 to 4 minutes more.
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14
Alternatively, broil slices on rack of broiler pan about 4 inches from heat, mayonnaise sides up, until golden brown, 3 to 4 minutes.
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15
Turn slices and brush tops with remaining mayonnaise.
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16
Broil until golden brown, 3 to 4 minutes more.
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17
Discard wooden picks
