Soak the minced onion in a bowl of cold water for 5-10 minutes, then drain.
In the bowl of a food processor, fitted with a blade, add the onion, lemon juice, honey, mustard, basil, garlic, salt, and pepper. Pulse a few times.
With the food processor running, slowly pour the oil in and process until emulsified. Alternately, if you don't have a food processor, add the ingredients to a bowl and slowly whisk in the oil until emulsified.
In another bowl, add the cranberry feta and mash with a fork breaking it into finer crumbles. Combine the feta crumbles with the vinaigrette, stir, and serve.