For the Pomegranate-Mint Agrodolce:
Heat the oil in a medium saucepan over medium-high heat. When it shimmers, add the shallots and garlic, stirring occasionally, until they are golden brown and tender, 10 to 15 minutes.
Stir in the vinegar, juice, brown sugar, the 1 teaspoon salt, the ½ teaspoon pepper, and bay leaf. Decrease the heat to low and simmer, stirring occasionally, until the shallots are very tender and the liquid is thick and syrupy, about 40 minutes. Stir in the pomegranate seeds, pine nuts, and mint. Season with salt and pepper.
For the broiled swordfish:
Heat the broiler to high and arrange a rack 6 to 8 inches from the heating element. Pat the fish dry, drizzle with oil, and sprinkle with salt and pepper. Place the fish under the broiler and cook until the fish is flaky, opaque throughout, and has a 135°F/57°C internal temperature, 8 to 12 minutes. Serve topped with agrodolce sauce.