Start by taking the partridges out of the fridge: Set them on a cutting board to come to room temperature while you make the tomato salad.
Make the salad by adding all the ingredients except for the basil to a large bowl. Toss to combine. You don't add the basil until later because it will discolor if left too long. Add it right before you serve.
Spatchcock the birds while the tomato salad marinates. Spatchcocking involves cutting out the backbones of the birds and flattening them so you can grill them faster. Once your birds are flattened and partially deboned, coat them with oil and salt them well.
Get your grill very hot and make sure the grates are clean. Right before you start cooking the birds, soak a paper towel in some vegetable oil grab it with tongs. Wipe down the grill grates. Lay the partridges down on the grill breast side up. If you have one, grab a brick and weigh down the partridges with it. This gives you better contact with the grill. Let the birds cook for at 8 to 10 minutes. You want to do most of the cooking on this side; it helps keep the breast meat moister.
Turn the partridges and grill them breast side down only until they get nice char marks, which typically takes about 2 to 3 minutes. If the birds have not yet cooked through -- some will, most won't -- put the partridges back breast side up for a few more minutes. If they arc upwards, put the brick back on them. When they are done, let the birds rest on a cutting board for 5 minutes before serving them with the tomato salad.