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Preheat the oven to 350 degrees F. Lightly grease the sides only of an 8-inch square cake pan.
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Stir together 2 tablespoons butter, light brown sugar, and 1 teaspoon vanilla in a small bowl. Spread mixture evenly over the bottom of the cake pan.
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Arrange the nectarine slices in rows over the brown sugar mixture in the pan.
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Whisk together the flour, cinnamon, baking powder, salt, and nutmeg in a bowl.
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Add the remaining 6 tablespoons butter and sugar to the bowl of a food processor fitted with a knife blade. Process until fluffy.
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Add the eggs and remaining 1 tablespoon vanilla. Process until combined, stopping to scrape down the sides of the bowl if needed.
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Add the flour mixture and milk. Pulse 2 to 3 times or until flour is moistened. The batter does not have to be completely smooth.
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Spoon and spread the batter evenly over the nectarine slices.
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Bake until the center springs back with gently pressed, about 40 to 45 minutes.
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Cool cake in the pan on a wire rack for 5 minutes.
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Use a knife to loosen the edges of the cake from the pan. Invert the cake to a serving plate. Serve warm or at room temperature.