Cucumber Salad with Sheep's Milk Feta and Lemon Oil

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1/4 cup olive oil, plus additional for oiling the grill

2 red onions, sliced into thin rounds 

1 cup lemon juice 

1 cup sugar 

1 cup red wine vinegar 

6 English cucumbers, half peeled and half with skin on, cut into 3/4-inch rings

1/2 to 1/4 large seedless watermelon, cut into 1-inch cubes 

1 cup sheep's milk feta crumbled in brine 

Salt and freshly ground black pepper

1 bunch fresh dill