In a large bowl of an electric mixer fitted with a dough hook, combine water, yeast and sugar until it proofs or bubbles, about five minutes. Add the flour, olive oil, and salt last and mix well for about three to four minutes. Cover with a kitchen towel and let rest for about half an hour in a warm area.

Meanwhile, preheat a small, deep pot filled with two inches of canola or peanut oil to approximately 350 degrees Fahrenheit. Add button mushrooms (cut to desired size) and shallots. It will take less than a minute for them to fry. Remove with a slotted spoon and drain on paper towel to absorb excess oil.

Preheat oven to 400 degrees Fahrenheit.

Once dough has risen, rub your palms with olive oil and press dough into a rimmed, greased pan about 13 inches by 21 inches, or two roasting pans. Smooth dough out so that it reaches the corners of the pan. Smear dough evenly with pizza sauce and sprinkle liberally with pizza cheese.

Bake in preheated oven for 22 to 25 minutes. Cut into desired-size pieces and sprinkle generously with mushrooms and shallots. Serve immediately.