Step 1Heat the oven to 350 degrees.
Step 2Cut the peaches lengthwise into one-half-inch slices. Place the slices in a medium bowl, drizzle with lemon juice, and set aside.
Step 3In a large 10- to 12-inch skillet, preferably cast iron, melt 3 tablespoons butter over medium heat. Add the brown sugar and stir until melted and smooth, then add the honey. Lower the heat and add the peach slices, and stir until they are glossy, slightly soft and well-coated with the honey-sugar mixture, about 2 minutes. Remove the pan from heat. If you are baking the cake in the skillet, spread the peach and sugar mixture in a thick layer to the edges and set the pan aside to cool. If you are baking the cake in a springform pan, scrape and spread the peach and caramel mixture into a 9-inch round pan (line the base with parchment paper, to about one-half inch up the sides of the pan).
Step 4In a medium bowl, sift together the flour, baking powder and salt. In a separate large bowl, using an electric mixer, cream the remaining butter and the sugar until it is pale and fluffy. Add the eggs, one egg at a time, and the orgeat syrup and beat for a minute until well combined. Next, working in batches and alternating with the rum, add the flour mixture to the butter mixture. Begin with a bit of the dry ingredients and stir until just combined, then add a splash of the rum and stir until just combined, and so on -- until all the ingredients are incorporated.
Step 5Dollop the batter into the prepared pan and smooth it as near as you can to the edges so that it covers the peaches. (Don't worry if it seems a little thick; it will fill in as it bakes.) Bake until the cake is golden and a knife inserted into the center of the cake comes out clean, about 45 minutes. (A springform cake will require additional time.)
Step 6Cool the cake in the pan for 10 minutes. If using a skillet, place a large plate over the top and carefully flip the pan so that the cake inverts onto the plate with the peaches on top. If using a springform pan, place a plate over the top of the pan, flip the pan over, release the spring mold, and peel back the parchment. Serve at room temperature.