Preheat the oven to 375℉.
Place the potatoes in a large pot and fill with water to cover the potatoes by 2 inches. Bring to a boil. Once boiling, add 1 tablespoon salt and continue to boil until potatoes are fork-tender, about 10 minutes.
Meanwhile, heat a large skillet over medium heat with 2 tablespoons butter. Add the onions and garlic and cook, stirring, until the onions are tender, 4 to 6 minutes. Set aside.
When potatoes are tender, strain and let the steam escape from the colander, about 5 minutes.
In a large mixing bowl, add the sour cream, chives, chicken broth, vinegar, black pepper, and remaining 2 tablespoons butter. Add the potatoes and mash to your desired consistency.
Add the onions, garlic, black pepper, and remaining 2 teaspoons salt. Stir to combine.
Transfer the potatoes to an oven-safe 9x13-inch baking dish, spread into an even layer, and sprinkle with cheddar cheese. Bake, uncovered, until the cheese is lightly browned, 10 to 15 minutes.
Top with remaining chives, serve, and enjoy!