The morphology of as-grown graphene in chemical vapor deposition (CVD) experiments is sensitive to the reaction environment. Understanding the mechanism of formation of different graphene morphologies is essential to achieve controlled graphene CVD growth. Here the growth and formation mechanism of adlayer graphene spirals are reported. An adlayer graphene spiral is formed by fast propagation of the tips of spiral arms along the edge of the first graphene layer. The driving force to form spirals is the limited availability of carbon diffusing from the Cu surface through the edge of the first graphene layer. In addition, it is found that graphene onions are formed by overlapping graphene spirals with clockwise and anticlockwise arms. Based on these features, a kinetic Monte Carlo (kMC) method is demonstrated using which all the observed graphene spiral structures are successfully reproduced at the atomic level. This study thus unravels the hither-to unresolved mechanism of graphene onion growth and paves the way to the controllable growth of few-layer graphene by increasing the carbon supply at the edge of the first layer graphene.