In a small bowl, combine the yeast with 2 tablespoons of water and a pinch of the sugar. Allow to bloom for about 10-15 minutes or until you see a little foam develop.
Combine the remaining water, sugar and salt with the flour in a stand mixer with a dough hook attachment. Mix for 1 minute and 30 seconds, scrape down then mix for 1 minute and 45 seconds and then again for 1 minute. Add oil and yeast mixture. Continue to mix for 4 more minutes or until the dough is smooth.
Drizzle two medium-size bowls with olive oil, lightly coating the inside walls and bottom of the bowl. Divide the dough in two equal portions, transfer each portion to a bowl, drizzle the top of each dough with a little more olive oil. Cover bowls with plastic wrap and keep in a warm area until it doubles in size, at least an hour.
For the sauce, in a medium-size saucepot over medium heat, heat the olive oil. Once hot, add the crushed red pepper flakes and cook for about 30 seconds; add the tomatoes, oregano, basil and salt and pepper to taste. Bring up to a bubble and let simmer 20 minutes.
For the peppers, in a medium-size skillet over medium-high heat, add the olive oil. Once hot, add the sliced sausage and cook until both sides are lightly browned. Remove from pan and reserve. Add the remaining olive oil to the skillet along with the sliced peppers. Cook, stirring every now and then, until lightly colored and tender, about 5 minutes. Add the vinegar and sugar; toss to combine. Remove from heat and reserve.
To assemble a pie, preheat oven to 400ºF. Lightly coat the inside of a 12-inch, cast-iron skillet with olive oil. Transfer the dough to the center of the skillet and spread out evenly towards the bottom edge and about 1 1/2 inches up the sides. Let the dough rise for 1 hour.
Top the pizza with a thin coat of the spicy tomato sauce, top that with the cooked peppers, spreading them out evenly across the pie. Evenly distribute the cheese and scatter the browned sausage on top of that. Transfer to the oven on the bottom rack and cook for about 30 minutes or until the crust is golden brown and crispy. Remove from oven and let rest 5 minutes before serving.