The giardiniera mayonnaise can be made up to 3 days in advance and refrigerated in an airtight container. You can also cook the veggie dogs on a grill. Preheat the grill to medium; place the dogs on the grill and cook, turning them every minute or two, until the hot dogs have dark grill marks and are heated through, 7 to 8 minutes. To grill the hot dog buns, simply follow the timing instructions above, but cover the grill to melt the cheese.