In a large saucepan over medium, melt the butter. Stir in the onions, thyme, bay and ¾ teaspoon salt. Cover and cook, stirring only occasionally at the start then more often once browning begins, until the onions are lightly golden and deep browning has formed on the bottom of the pot, about 45 minutes.

Meanwhile, heat the oven to 375°F with a rack in the middle position. Place four 12-ounce oven-safe bowls on a rimmed baking sheet and sprinkle ¼ cup cheese in each; set aside. Lightly flour a counter and unfold the thawed puff pastry onto it; dust the surface with flour. Using a paring knife, cut out four rounds from the pastry, each about ½ inch larger than the diameter of the bowls. Stack the pastry rounds on a large plate and refrigerate, uncovered, until ready to use.

Uncover the pot and stir in the cornstarch. Add the vermouth and cook, scraping up the browned bits, until the mixture is homogeneous and thickened, about 2 minutes. Stir in the broth, 1 cup water and ¼ teaspoon pepper. Bring to a simmer and cook, uncovered and stirring occasionally, until lightly thickened, about 15 minutes. Off heat, remove and discard the bay. Taste and season with salt and pepper.

Divide the soup evenly among the bowls. Working one at a time, brush a thin layer of egg yolk around the rim, then place a pastry round on top; gently press around the rim of the bowl to secure the pastry. Once all four bowls are lidded, brush the pastry rounds with the remaining yolk and sprinkle with the remaining cheese. Bake until the pastry is puffed and golden brown, 25 to 30 minutes. Serve right away.