Coat the bottom of the pan with olive oil, or a mixture of olive oil and butter (about 1 teaspoon per onion). Heat the pan on medium high heat until the oil is shimmering. Add the onion slices and stir to coat the onions with the oil. Spread the onions out evenly over the pan and let cook, stirring occasionally until their caramelized. This takes about 35 minutes. Season with salt. Then allow the onions to cool to room temperature.
Put the goat's cheese in a small bowl and add black pepper to taste. Stir the goat's cheese until it is easily spreadable and allow the puff pastry slices to thaw.
Cut the baking paper to the size of the puff pastry slice, place the puff pastry slice crosswise on the baking paper and press it into the refractory dish.
Divide the goat's cheese mixture with the onions over the 10 bowls, sprinkle with thyme and bake the cakes in the Airfryer basket at 160°C for 20 to 25 minutes.
Serve hot or at room temperature.