In a large bowl, combine the warm water and yeast and let it sit for about 5 minutes until it starts getting foamy on top. Add the 3 T. of olive oil and salt and whisk to combine. Stir in the flour and dump onto a lightly floured surface. Knead dough just until it comes together in a ball. Return to the bowl. Pour a bit more olive oil on top of the dough ball and turn to coat. Cover the bowl with a clean dish towel and put in a warm place to rise for about an hour, or until doubled in size.
In the meantime, preheat oven to 450 degrees. Wash the cauliflower and discard the stems. Chop the florets in bite-sized pieces. Chop the zucchini into bite-sized pieces. Toss the vegetables to evenly coat with the remaining 2 T. of olive oil, curry powder and salt. Spread evenly on a sheet pan and roast for about 15 minutes, or until cauliflower is fork-tender. Remove from oven and allow to cool while dough is rising.
Pulse the mango chutney in a small food processor (or with an immersion blender) until it's smooth. Stir in the fresh chopped cilantro.
Place pizza stone on rack in lower third of your oven. Increase oven temperature to 525 degrees.
Once the dough is ready, spread evenly onto a piece of parchment paper (about a 12-inch circle). Top the dough evenly with the mango chutney sauce.
Top the sauce with the curried, roasted vegetables. Top the vegetables with the mozzarella and goat cheeses.
Using a pizza peel, transfer the pizza (on the parchment paper) to the pizza stone. Bake the pizza for 10 - 12 minutes, or until crust is browned and cheese is nice and bubbly.