Heat oven to 375°F. In small bowl, mix artichokes, 1/2 cup Parmesan cheese and the mayonnaise.
If using crescent rolls: Unroll dough; separate into 2 long rectangles. Place crosswise on opposite ends of ungreased large cookie sheet. Press each into 11x4 1/2-inch rectangle, firmly pressing perforations to seal. If using dough sheet: Unroll dough; cut into 2 long rectangles. Place crosswise on opposite ends of ungreased large cookie sheet. Press each into 11x4 1/2-inch rectangle.
Spoon half of artichoke mixture in 1 1/2-inch-wide strip lengthwise down center of each rectangle. Top each evenly with pimiento.
On long sides of each rectangle, make 15 cuts about 3/4 inch apart almost to edge of artichoke mixture. For braided appearance, fold strips of dough at an angle halfway across artichoke mixture with ends of strips slightly overlapping, alternating from side to side; pinch to seal.
Bring ends of each braid together to form ring; press ends together to seal. Carefully brush tops of rings with egg; sprinkle with 1 teaspoon Parmesan cheese.
Bake 13 to 18 minutes or until deep golden brown. Cool 5 minutes. Carefully loosen braids from cookie sheet with wide spatula; slide onto serving plate. Cut between strips into thin slices; serve warm.