Thaw bread according to package directions. Heat oven to 425 degrees F. Line a large baking sheet with foil; grease foil. Set baking sheet aside.
Drain artichoke hearts, reserving marinade. Quarter artichoke hearts (if necessary, cut larger pieces of artichoke hearts in half to get total of 12 pieces); set aside. Place red onion wedges and eggplant slices on foil-lined baking sheet. Brush with some of the reserved artichoke marinade (discard any remaining marinade). Bake red onion and eggplant in oven for 10 minutes (do not turn). Remove from oven; set aside.
Meanwhile, cut the thawed bread dough into 12 equal pieces; form each piece into a ball. On a lightly floured surface, flatten each ball and roll out to a 4-inch circle. Place dough circles on lightly greased large cookie sheets. Prick circles generously with fork.
Sprinkle 1 cup of the Gouda cheese evenly over dough circles. Arrange one half-slice of eggplant, one red onion wedge, one artichoke quarter, and one slice of tomato on each dough circle. Sprinkle green onion and rosemary over circles. Top circles with remaining Gouda cheese. Bake about 10 minutes or until edges are lightly browned. Remove from oven; sprinkle with snipped fresh basil. Makes 12 servings.