Halve turnips (leave whole if tiny) and put in a 12 - inch heavy skillet along with water, mirin,
remaining tablespoon butter, and 1/8 teaspoon salt.
Stir in
the remaining tablespoon butter at the end, garnish with parsley, and add the lemon juice.
Remove from the oven and add
the remaining tablespoon butter to the top of the potatoes.
In the same skillet, melt 3
remaining tablespoons butter, then stir in lemon juice and zest, Dijon mustard, hot sauce, cayenne, paprika, and chicken broth.
Stir in vinegar,
remaining tablespoon butter, and salt and pepper to taste.