In a 12" cast-iron skillet over medium heat, cook and scramble sausage until no longer pink. Remove to a bowl. Set aside.
Add butter to the cast-iron skillet over medium heat. Saute mushrooms, garlic, and onion until mushrooms are tender and onion is translucent. Remove from skillet. Set aside.
Preheat the oven to 450 degrees. To a bowl, add warm water. Add 1 tablespoon of sugar and yeast. Stir it. Let it sit 5 minutes until foamy.
Add flour, rosemary, 1 teaspoon sugar, salt, and 1 1/2 tablespoons of oil to the yeast. Stir to combine to form a dough. Turn onto a lightly floured surface and knead 8-10 times-adding more flour to the dough if it becomes too sticky. Place the dough back into the bowl and let the dough rest for about 10 minutes while you prepare the rest of the toppings. (grate the cheese, chop the spinach)
Roll the dough into a 12" circle.
Brush the bottom of the 12-in. cast-iron skillet with the remaining oil—transfer dough to the skillet and bake for 5 minutes. Remove the skillet from the oven. Spread crust with pizza sauce; top with mushroom mixture, sausage, black olives, spinach, and cheese.
Place skillet on middle rack in the oven. Bake for about 15-20 minutes or until the crust looks crispy and golden brown.