1 large yellow onion, quartered and sliced crosswise into ½- to ¾-inch-thick pieces
1 tablespoon kosher salt, divided, plus extra if needed
2 cups vegetable or chicken broth, divided
12 large garlic cloves, peeled, plus 2 garlic cloves, very finely chopped
6 medium carrots, peeled and sliced on a bias into ½-inch pieces
1 medium rutabaga, peeled and chopped into ½-inch cubes (about 4 cups)
1¼ teaspoons freshly ground black pepper, divided
¼ cup extra-virgin olive oil, divided
5 fresh thyme sprigs plus 2 teaspoons finely chopped fresh thyme, divided
2 tablespoons unsalted butter, at room temperature
3 tablespoons all-purpose flour
½ medium head Napa cabbage, sliced crosswise into ½-inch-thick pieces (about 6 cups)
8 medium radishes, trimmed and quartered
1 large egg
1 sheet frozen puff pastry, thawed
1. Adjust an oven rack to the uppermost position (about 3 inches from the broiler element) and preheat the broiler to high. On a rimmed baking sheet, add the onion in a single layer. Broil the onions until they are charred, rotating the baking sheet midway through cooking, 6 to 8 minutes (watch the onions closely, as broiler intensities vary). Remove the baking sheet from the oven, sprinkle the onions with ½ teaspoon of the salt and pour 1 cup of vegetable broth over the onions, then transfer the mixture to a medium bowl.
2. Reduce the oven temperature to 425°. To the baking sheet used for the onions, add the 12 whole garlic cloves, the carrots, rutabaga, 1 teaspoon of salt and ½ teaspoon of pepper. Drizzle with 3 tablespoons of the oil and toss to combine. Place the thyme sprigs on top. Tightly cover the baking sheet with aluminum foil and roast the vegetables until tender, about 30 minutes. Remove from the oven and discard the foil. Use tongs to gather all of the roasted garlic cloves, then use a fork to smash them into a paste.
3. In a small bowl, mash together the butter and flour. Heat a large, deep, oven-safe skillet (cast iron works well) over medium-high heat for 2 minutes, then add the remaining 1 tablespoon of olive oil. Stir in about half of the cabbage and cook, stirring occasionally, until it wilts, 1 to 2 minutes. Stir in the remaining cabbage along with the radishes, the 2 cloves chopped garlic and the chopped thyme, and cook until the cabbage is wilted completely and the garlic is fragrant, 2 to 3 minutes longer.
4. Stir in the remaining 1½ teaspoons of salt and ¾ teaspoon of pepper, then pour in the remaining 1 cup of vegetable broth. Add the roasted vegetables, mashed roasted garlic and onion-broth mixture and bring to a simmer. Reduce the heat to medium and stir in the butter-flour mixture, cooking until the sauce thickens slightly, 2 to 3 minutes. Turn off the heat, taste and adjust with more salt, if needed.
5. Whisk together the egg with a pinch of kosher salt. Place the puff pastry on a cutting board and use a sharp knife to lightly mark the dough with a crosshatch pattern, making sure you don't slice all the way through the dough. If your skillet isn't oven-safe, transfer the vegetables to a baking dish. Lay the piece of dough on top of the vegetables in the skillet and lightly brush with the beaten egg. Sprinkle with flaky salt and place the skillet in the oven. Bake until the pastry is golden-brown, 18 to 20 minutes. Remove from the oven and cool for 10 minutes before serving.