Position a rack in the top third of the oven and heat the oven to 425°F.
Heat the olive oil in a 10-inch skillet over medium-high heat. When the oil is hot, add the peppers and a generous pinch of salt. Cook, stirring often, until the peppers soften and brown slightly, about 5 minutes. Set aside to cool.
Put the cornstarch in a medium bowl. Whisking steadily, slowly pour in 1/2 cup of the milk, mixing until quite smooth. Whisk in the whole eggs and egg yolks, mixing again until smooth, and then gradually whisk in the rest of the milk, the cream, salt, and cayenne.
Butter or oil a 2-quart, 7x11-inch Pyrex dish or an 8x11-inch ceramic gratin dish. Spread the peppers over the bottom of the pan. Scatter on the goat cheese and thyme.
Whisk the batter in case it has settled and then pour it into the baking dish. Bake until the top is deep golden and the custard is set (insert a pick in the center to check; it should be softly set, neither liquid nor firm), 25 to 30 minutes. Let stand for 30 minutes before serving.
The batter can be made up to one day ahead and refrigerated, covered.