5 1/2 ounces (1 stick plus 3 tablespoons) unsalted butter, softened
1 cup sugar
3 medium ripe peaches (about 1 1/4 pounds), skins on, pitted, and cut into 3/4-inch wedges
1 cup coarse yellow cornmeal or polenta
3/4 cup all-purpose flour
1 teaspoon baking powder
2 teaspoons chopped fresh lavender, or 1 1/2 teaspoons dried lavender
1 1/4 teaspoons coarse salt
3 large eggs
1/2 teaspoon pure vanilla extract
1/2 cup heavy cream
Preheat oven to 350 degrees. Melt 3 tablespoons butter in a 10-inch cast-iron skillet over medium heat, using a pastry brush to coat sides with butter as it melts. Sprinkle 1/4 cup sugar evenly over bottom of skillet, and cook until sugar starts to bubble and turn golden brown, about 3 minutes. Arrange peaches in a circle at edge of skillet, on top of sugar. Arrange the remaining wedges in the center to fill. Reduce heat to low, and cook until juices are bubbling and peaches begin to soften, 10 to 12 minutes. Remove from heat.
Whisk cornmeal, flour, baking powder, lavender, and salt in a medium bowl. Beat remaining stick of butter and 3/4 cup sugar with a mixer on high speed, until pale and fluffy, about 3 minutes. Reduce speed to medium. Add eggs, 1 at a time, beating well after each addition and scraping down sides of bowl. Mix in vanilla and cream. Reduce speed to low, and beat in cornmeal mixture in 2 additions.
Drop large spoonfuls of batter over peaches, and spread evenly using an offset spatula. Bake until golden brown and a tester inserted in the center comes out clean, 20 to 22 minutes. Transfer skillet to a wire rack, and let stand for 10 minutes. Run a knife or spatula around edge of cake. Quickly invert cake onto a cutting board. Tap bottom of skillet to release peaches, and carefully remove skillet. Reposition peach slices on top of cake. Let cool slightly before serving.
Perfectly ripe peaches, available from May to October, will feel heavy for their size, yield slightly to gentle pressure, and emit a sweet, heady perfume. Avoid any fruit tinged with green, as it won't ripen. Ready-to-eat peaches last 3 to 5 days refrigerated; firmer ones will soften when kept in a brown paper bag on the counter for up to 3 days.