Place the peeled and quartered potatoes in a large pot. Add cold water to cover them by about an inch. Stir in 1 teaspoon of salt.
Bring the water to a boil over high heat, then reduce the heat to maintain a steady simmer. Cook, uncovered, for 15-20 minutes, or until the potatoes are easily pierced with a fork.
While the potatoes are cooking, melt the butter in a light-colored skillet over medium heat. Continue to cook, swirling the pan occasionally, for 5-7 minutes. The butter will foam, then the milk solids will turn golden brown and smell nutty.
During the last minute of browning, add the minced garlic to the butter and stir constantly until fragrant. Remove the skillet from the heat.
Drain the cooked potatoes thoroughly in a colander. Return the empty pot to the stove over low heat for a minute to evaporate any remaining moisture, then return the potatoes to the pot.
Mash the potatoes with a potato masher until they reach your desired consistency.
Gradually pour in the warm milk and the browned butter mixture, stirring gently until everything is combined and the potatoes are creamy.
Stir in the remaining 1 teaspoon of salt and the black pepper. Taste and add more seasoning if needed.
Transfer the mashed potatoes to a serving bowl. Garnish with fresh chives, if using, and serve immediately.