Preheat the oven to 350 degrees F. Lightly butter a 9-inch-round cake pan. Melt 1 tablespoon butter in a medium skillet over medium heat. Add 3/4 cup sugar and 1 tablespoon water and cook, stirring occasionally, until smooth and deep golden brown, 8 to 11 minutes. Pour the mixture into the prepared pan and tilt to coat the bottom. Arrange the peach wedges snugly in the bottom of the pan in a single layer, cutting to fit if needed.
Combine the flour, almonds, baking powder and salt in a food processor and pulse until the almonds are finely ground.
Beat the remaining 7 tablespoons butter and 3/4 cup sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 4 minutes. Add the eggs, one at a time, beating well after each addition. Beat in the vanilla and almond extracts. Beat in the flour mixture in 3 batches, alternating with the milk and beginning and ending with the flour mixture; beat in each addition until just incorporated.
Spread the batter evenly in the prepared pan and bake until golden and a toothpick inserted into the center comes out clean, about 45 minutes. Transfer to a rack and let cool 30 minutes. Invert onto a plate and let cool completely. Serve with ice cream.