Combine 1-1/4 cups of the flour, the yeast, rosemary, salt, and baking soda in a large mixing bowl. Add water and the 2 tablespoons oil. Beat with electric mixer on low to medium speed for 30 seconds, scraping bowl. Beat on high speed 3 minutes. Using a spoon, stir in the 1/4 cup cheese and as much remaining flour as you can.
Turn dough out onto a lightly floured surface. Knead in enough remaining flour to make a stiff dough that is smooth and elastic (8 to 10 minutes). Shape dough into a ball. Place in a lightly greased bowl; turn once to grease surface. Cover and let rise in a warm place until double (about 1 hour).
Meanwhile, to roast sweet peppers, quarter the peppers; remove and discard stems, seeds, and membranes. Place pepper quarters, cut sides down, on a baking sheet lined with foil. Bake in a 425 degrees F oven for 20 to 25 minutes or until pepper skins are blistered and dark. Remove from oven; reduce oven temperature to 375 degrees F. Immediately wrap peppers in the foil. Let stand about 30 minutes to steam so skins peel away easily. Remove and discard skin from peppers. Cut peppers into thin strips.
Grease 2 baking sheets. Punch down dough. Turn out onto a lightly floured surface. Divide in half. Shape each half into a ball and place on a prepared baking sheet. Cover and let rest for 10 minutes. Using your hands, flatten each ball to about 12 inches in diameter. Cover and let rise until nearly double (about 20 minutes). With flour-dusted fingers, press fingers into dough making 1/2-inch-deep indentations. Repeat to cover dough, spacing indentations 1 to 2 inches apart.
Brush dough with the 4 teaspoons oil. Sprinkle with ground pepper. Top with pepper strips, pine nuts, and the 1/2 cup cheese. Bake in a 375 degrees F oven about 20 minutes or until golden. Transfer to wire racks. Cool completely. Tear or cut into wedges. Makes 16 servings.