For the dough, dissolve yeast in warm water. Add oil followed by flour. Mix then add salt.
Place dough in an oiled bowl and cover with plastic wrap. Let proof in a warm area for 3 hours.
Cut dough into 90-gram portions and shape into balls.
Using your fingers flatten out the dough into 4.5- to 5-inch rounds.
Rub with garlic then equally distribute the smoked and fresh mozzarella cheeses over dough, leaving ¼ inch around the edges.
Season with salt and pepper. Distribute salami over cheese then sprinkle with basil.
Starting at the top and bottom of the dough—the 12:00 and 6:00 positions—bring the dough to the center and hold it together. Bring the dough up from 3:00 and 9:00 positions and lightly pinch together in the center. Bring the remaining dough up and pinch it together then place the seam side down on a sheet tray.
Brush with oil and let proof for 1 hour.
Preheat oven to 375 °F. Bake stromboli until golden, about 40 minutes. Let rest a few minutes before serving.