Dissolve the sugar in the warm water and add the yeast.  Allow it to proof for 10-15 minutes.  You should see the yeast bubble up and form a layer of foam on top of the water.  

Place 2 cups flour in a mixing bowl and add a pinch of salt, 1-2 Tbsp olive oil, and the yeast/water mixture.  Using the dough hook attachment, mix until combined.  Add in the 3rd cup of flour and mix until combined, then add as much of the 4th cup as you need until the dough no longer sticks on your finger when you press on it.  Turn the dough out onto a floured surface and knead until smooth and elastic.  Form into a ball and place into a lightly oiled bowl, covering with a kitchen towel.  Let the dough rise for a few hours, then separate into 4 equal portions.  Roll each portion out into a pizza crust.

While the dough is rising, add olive oil to a pan and cook the mushrooms, peppers, and onion.  Preheat the oven to 400.  Place the broccolini on a sheet pan and drizzle with olive oil.  Roast for about 20 minutes, turning, until lightly browned. Reserve the cooked veggies in a bowl.

After rolling out the pizza crust, top with sauce, then cheese, then veggies.  Bake in a 450 degree oven for 10 minutes.