Melt the butter in a large saucepan over medium heat. Add the root vegetables and cook, stirring occasionally, until the butter is browned and the vegetables are dark brown and caramelized, 10 to 12 minutes. (If after 4 minutes the vegetables have not started to brown, increase the heat to medium-high.)
Add the potatoes, broth, and 3/4 teaspoon salt and stir to combine. Cook, covered, over low heat (the broth should simmer gently; do not boil), stirring occasionally, until the potatoes fall apart easily when poked with a fork and all the liquid has been absorbed, 25 to 30 minutes. (If the liquid does not gently simmer after a few minutes, increase the heat to medium-low.) Remove the pan from the heat; remove the lid and allow the steam to escape for 2 minutes.
Gently mash the potatoes and root vegetables in the saucepan with a potato masher (do not mash vigorously). Gently fold in the half-and-half and chives. Season with salt and pepper to taste and serve the mashed potatoes immediately.