How to Make It
In a large skillet, cook the honey over high heat until darkened slightly, about 1 minute. Add the plums cut sides down and cook over moderate heat until they release their juices, about 3 minutes. Increase the heat to moderately high and cook until the plums are browned, about 3 minutes longer. Transfer the plums to a plate.
Remove the skillet from the heat and add the sherry vinegar. Scrape the honey from the bottom of the skillet and stir together with the vinegar until blended; transfer to a small bowl. Whisk in 1 tablespoon of the lemon juice and the 2 tablespoons of olive oil and season the dressing with salt and black pepper.
Light a grill. Drizzle the cheese with olive oil and season with salt and black pepper. Grill over high heat until grill marks form, about 1 minute per side. Transfer the cheese to a plate. Drizzle with olive oil and the remaining 1 tablespoon of lemon juice, then sprinkle with the oregano. Cut each cheese round into 3 triangles.
In a large bowl, toss the arugula with 3 tablespoons of the dressing. Mound the salad on plates. Arrange 2 plum halves and a cheese triangle on each plate. Drizzle the remaining dressing over the plums, sprinkle with the basil and serve.