Preheat the oven to 425 degrees F.
Heat the olive oil in a large, high-sided ovenproof skillet or braiser over medium-high heat. Add the carrots, scallion whites, garlic and ginger and cook, stirring occasionally, until the vegetables soften, 3 to 4 minutes.
Whisk the soy sauce, brown sugar, gochujang paste, molasses, tomato paste, Worcestershire and ½ cup water together in a medium bowl. Pour the sauce into the skillet and stir to combine with the vegetables. Bring to a simmer, stirring occasionally, and simmer 2 to 3 minutes.
Fold in the kimchi, chickpeas and ½ teaspoon salt and simmer 2 to 3 minutes longer.
Transfer the skillet to the oven and bake until the sauce has thickened and darkened slightly, 12 to 15 minutes.
Remove from the oven and stir in the butter until melted. Garnish with scallion greens and serve with toasted sourdough if using.
