Dissolve yeast in warm water; stir in sugar. Combine flours and salt, and gradually add the dissolved yeast and 1/4 cup of the oil. Knead until the texture is smooth. Put in a large bowl, cover with plastic wrap, and let rise until triple in bulk (2-3 hours).
Meanwhile, prepare the fillings. Heat 1/4 cup of the oil in a saute pan, add 2 t garlic, and cook for 30 seconds (without browning.) Stir in the tomato sauce and paste, simmer until thickened. Stir in basil and oregano, set aside to cool.
Head 2 T of the oil and saute the mushrooms until lightly browned and the liquid is evaporated. Season to taste, and set aside to cool. Remove and discard the casings from sausage, crumble and add the sausage to the pan along with fennel. Cook thoroughly, remove and cool. Heat the butter and 2 T of the oil; at 1 t of garlic and stir for 30 seconds. Stir in the broccoli until coated well and any liquid is evaporated. Season to taste; set aside.
When the dough has risen, punch down. Cut off about 2/5 of it and set aside. Grease a 14 x 1 1/2" deep-dish pizza pan with the shortening. On a floured board, rool out 3/5 of the dough to a 20" circle. Fit to the pan, letting the exess dough hang over the side. Brush the dough with 1 T of the oil; sprinkle with salt. Sprinkle 1 c of the mozzarella over the dough. Spread the tomato sauce across the cheese, spread the mushrooms over the tomatoes, and cover with 1 c of mozzarella. Roll out the remaining dough to approximately a 14" circle. Brush the sides of the dough inside the pan with water.
Fit the 14" round into the pan. Press edges (pull if necessary) against the moustened dough to seal it. Trim the overhanging dough to 1/2" and wet it again. Fold inward and crimp to form a raised rim around the pan edge. Cut a steam vent in the top layer of the dough, and brush with 1 T of the oil. Spread the sausage across the dough and cover with the broccoli. Combine the remaining cheeses and sprinkle across the broccoli; drizzle with 1/4 c of oil. Bake in a preheated 425 degree oven for 30-40 minutes. Freezes well.
Ideas: put the broccoli and sausage on the lower layer, and the tomatoes and mushrooms on the top layer. This way the broccoli stays moist.
For a vegetarian pizza, replace the sausage with 4 c sliced onions, lightly browned in 2 T oil. Or, you can keep the sausage and add the onions anyway.