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1
Put oven rack in middle position and preheat oven to 400F.
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2
Melt 2 teaspoons butter, then brush muffin cups with some of it.
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3
Chill muffin tin 2 minutes to set butter.
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4
Line bottom of each muffin cup with a round of parchment paper, then brush cups and parchment with remaining melted butter and chill 2 minutes more.
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5
Divide 1 teaspoon flour among muffin cups and shake to coat, knocking out excess flour.
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6
Cut an X in bottom of each peach and immerse peaches in a large pot of boiling water until skins loosen, about 15 seconds.
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7
Transfer peaches with a slotted spoon to a large bowl of ice and cold water to cool.
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8
Peel peaches, then pit and halve lengthwise.
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9
Cut peach halves lengthwise into 1/2-inch-thick slices, then cut slices crosswise into 1/2-inch pieces.
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10
Melt remaining tablespoon butter in a 10-inch heavy skillet over moderate heat, then stir in 3 tablespoons brown sugar and cook, stirring occasionally, until sugar is melted and bubbling, about 2 minutes.
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11
Carefully add a scant 2 cups peaches (caramel will splatter and sugar will seize), reserving remaining 1/3 cup peaches, then reduce heat and simmer, stirring occasionally, until peaches are tender and caramel is syrupy, 6 to 8 minutes.
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12
Spoon cooked peaches and caramel evenly into muffin cups.
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13
Puree reserved peaches in a blender until smooth.
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14
Beat together eggs, salt, almond extract, and remaining 6 tablespoons brown sugar in a bowl with an electric mixer at high speed until mixture is creamy and tripled in volume, 3 to 5 minutes in a stand mixer or 6 to 8 minutes with a handheld.
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15
Reduce speed to low, then add pureed peaches and mix until just combined.
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16
Sift remaining 6 tablespoons flour evenly over egg mixture and fold in gently but thoroughly until just combined.
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17
Spoon batter over caramelized peaches and bake until golden brown and a wooden pick or skewer inserted in center of a cake comes out clean, 10 to 12 minutes.
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18
Cool in muffin tin on a rack 2 to 3 minutes, then invert a platter over muffin tin and flip cakes onto platter while still warm.
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19
Reposition any peaches that have stuck to parchment and serve cakes warm or at room temperature.
