Finely chop mushrooms in a food processor then scrape them out into a clean, cotton towel, twist the towel around mushrooms, and wring out as much liquid as you can over the sink.
Heat a large (10-inch) non-stick skillet over a burner set between medium and medium-high then add 1 tablespoon butter and swirl to melt and avoid burning. Add mushrooms, shallots, a pinch of salt, a pinch of black pepper, and thyme. Cook, stirring frequently until mushrooms appear dry and are beginning to brown; about 5 minutes, stir in remaining tablespoon of butter, and, when melted, the sherry or wine. Cook this mixture, stirring frequently, until the vermouth has evaporated. Remove from heat, cool, and fold in truffle oil.
Preheat oven to 400 degrees, roll out pizza dough until it is 1/8″, place on an oiled baking sheet, and bake until just golden. Remove from oven and spread the mushroom mix evenly, then top with grated fontina. Place back in the oven and cook until cheese is melted.
While pizza is cooking, cut chives into batons and mix with Frisee, zest, and juice of 1 lemon and the olive oil. When the pizza is done, top with this Frisee mix, slice, and serve.