In a large Dutch oven, combine the potatoes, 1½ teaspoon salts, bay leaves and garlic. Add enough cold water to cover by 2 inches then set over high and bring to a boil. Reduce to medium, then cook until the potatoes are tender, 20 to 25 minutes. Drain the potatoes, then return to the pot. Discard the bay leaves, then return the potatoes to the pot.

In a small saucepan over medium-low, melt 6 tablespoons of the butter. Add the melted butter to the potatoes. Using a potato masher, mash until smooth. Stir in the half-and-half, horseradish and 3 tablespoons of the reserved horseradish liquid. Taste, then season with salt. Cover and set over low heat to keep warm.

Return the saucepan to medium and add the remaining 4 tablespoons butter and the caraway and mustard seeds. Cook, gently swirling the pan, until butter is browned and seeds are fragrant and toasted, 2 to 3 minutes. Strain into a small glass measuring cup. Transfer the potatoes to a serving dish, then drizzle with the butter and garnish with chives.