I live in the NW corner of South Carolina and the soil is clay. I have better luck growing large size onions from plants than I do with the sets; however, it might be that the season has something to do with it since I use plants in the spring and sets in the fall. I begin like everyone by planting the onion plant about 1 inch in the soil but when the bulb starts forming I pull away the soil from around the bulb so that the bulb is completely out of the soil and only the roots are in the soil. This process allows the bulb to expand without the confinement of the soil.
I have stopped planting onions in the fall since over the winter here the weather changes from warm to cold repeatedly and the bulbs get tricked into thinking it has been two years and bolt. This pretty much ruins the onion for me.