"In the style of Piedmont." As with many regions of Italy, the style is rather loose and means mainly that local produce will be involved. It could mean a dish of veal escalopes with Fontina cheese or truffle or a meat sauce with truffles sometimes with nutmeg.
"In the style of the pizza-maker." Cooked with tomatoes, herbs and garlic, perhaps with capers and oregano, sometimes with anchovies, mozzarella, often with slices of beef. Whatever takes the pizza-maker's fancy.
Alla povera could refer to almost any dish which is cooked in a simple style with locally available cheap ingredients. This used to be scorned but is now very sought after.