For the pizza dough: Place a pizza stone in the oven and preheat to 425 degrees F.
In a stand mixer, combine the flour, yeast, salt and sugar. With the mixer running, add the water and 2 tablespoons of the oil and mix until the dough forms a ball. If the dough is sticky, add additional flour, 1 tablespoon at a time, until the dough comes together in a solid ball; if the dough is too dry, add additional water, 1 tablespoon at a time.
Scrape the dough onto a lightly floured work surface and gently knead into a smooth, firm ball. Grease a large bowl with the remaining 2 teaspoons olive oil, add the dough and cover the bowl with plastic wrap. Let rise in a warm area until doubled in size, about 1 hour.
Preheat a grill pan over medium-high heat.
Turn the dough out onto a lightly floured work surface and divide it into two pieces. Cover with a clean kitchen towel or plastic wrap and leave to rest for 10 minutes. Roll the dough out into 6-inch rounds.
One at a time, place the dough rounds on the preheated grill pan and grill until cooked and grill marks appear on the first side, about 2 minutes. Flip and do the same on the reverse side.
For the topping: Saute the sliced sausage until golden brown.
Scatter the stracchino, browned sausage and some peppers over the grilled pizzas. Transfer the pizzas to the preheated pizza stone and bake until the cheese is bubbly and melted, 5 to 7 minutes. Remove from oven and top each pizza with a fried egg and some arugula, fennel pollen and grated pecorino. Slice and serve.