In a bowl, blend the butter with the mustard, the shallots and 1 teaspoon of the garlic. Season with salt and pepper.
In a stockpot of boiling water, cook the turnip greens until tender, about 10 minutes. Drain and let cool, then squeeze the greens dry and coarsely chop them.
In a large enameled cast-iron casserole, heat the olive oil. Add 2 tablespoons of the mustard butter and the onions and cook over moderately low heat until softened, about 10 minutes. Add the anchovies and the remaining garlic and cook until the garlic softens. Add the turnip greens and cook, stirring, for 10 minutes.
Bring a large pot of salted water to a boil. Add the turnips and cook until tender, about 12 minutes. Drain and return the turnips to the pot; shake over high heat to dry them out. Remove from the heat and stir in 3 tablespoons of the mustard butter. Season with salt and pepper.
Reheat the greens. Stir in the piquillos and the remaining 3 tablespoons of mustard butter. Season with salt and pepper. Spoon the greens into a warmed bowl. Top with the turnips and serve.
Notes: Wood-roasted piquillo peppers from Spain have an intense sweet-smoky flavor. Two peeled and seeded roasted red bell peppers can be substituted.