Mix butter and miso.
Cut turnips from their greens, leaving a bit of stem attached; then trim roots. Leave smallest turnips whole; cut larger turnips into halves or quarters. Slice greens (including stems) crosswise into 1-in. pieces.
Put turnips in a 6- to 8-qt. pot and season with salt and pepper to taste. Heat over medium-high heat, covered (they should release quite a bit of liquid; if they're on the dry side, add 1/4 to 1/2 cup water). Reduce heat to a simmer and cook 5 minutes more.
Add turnip greens, cover, and cook until greens are just wilted, about 4 minutes. Uncover and cook until both greens and turnips are nearly tender, about 2 minutes.
Stir in miso butter and maple syrup, then cook, uncovered and stirring often, over medium-high heat until everything is well incorporated, and greens and turnips are tender, 5 minutes. Transfer to a deep platter or large bowl and drizzle broth around turnips.