Preheat the oven to 160C or 350F. Place a piece of parchment paper or greaseproof into an 8-or 9-inch cast iron pan and grease with butter.
In a bowl, whisk together the flour, cardamom, baking powder, and salt. In a separate large bowl, beat the eggs until foamy. Whisk in the sugar, gin, and vanilla extract. Pour in the buttermilk and whisk to combine.
Peel, halve, and core 3 of the pears, then cut them into 1/2-inch-thick cubes. Reserve the remaining pear (unpeeled, and cut into 1/4-inch slices) for the top.
Add half of the flour mixture to the wet ingredients, stirring until just combined, then gently fold in half of the melted butter. Repeat with the remaining flour and butter. Gently fold in the cubed pears, reserving the slices. Add the leaves from 5 thyme leaves. Transfer the batter to the pan and arrange the pear slices in a circular pattern on top of the batter. Sprinkle with demerara sugar and the remaining thyme leaves. Bake until the cake turns a deep golden brown and a cake tester inserted into the centre comes out clean, about 55-65 minutes. Transfer the skillet to a cooling rack and let sit for 5 minutes. The cake may be served warm or at room temperature, directly from the skillet; whichever you choose, be sure to dust with confectioners sugar before serving.