Simmer water and butter until melted. Add flour and cook, stirring until dough pulls off walls of pot. Add dough to mixer fitted with a dough hook. Mix batter on low speed, and add eggs one at a time. Add cheese to melt.
Fill a pot with boiling water. Put dough in a pastry bag fitted with 1/2-inch diameter tip and pipe the dough into boiling water while cutting at 1-inch intervals to form dumplings. Remove dumplings when they float to the top and dry on clean parchment paper. Gnocchi are now ready to cook or freeze.
Heat a nonstick pan over medium heat and prepare to sauté all mushrooms in 4 batches. For each batch, add 1 tablespoon butter to the pan, then sauté mushrooms with 1 tablespoon shallots, 1 teaspoon lemon thyme, salt and pepper. As batches are finished, keep warm on clean parchment paper.
Add 1 tablespoon butter to a clean pan. Add salsify and sauté until just tender (5-7 minutes). Season with salt and pepper to taste, and reserve with mushrooms.
Heat a nonstick pan over medium-high. Heat 1 tablespoon butter until foamy and lightly browned. Toss in a single layer of gnocchi and let them brown for 1-2 minutes. (CHEF’S NOTE: Do not shake the pan while gnocchi are browning, but do shake the pan to turn gnocchi once they are brown.) Add mushrooms, salsify, remaining herbs, and Parmesan. Toss, plate and serve warm.