Preheat the oven to 400 degrees. Brush each piece of bread with olive oil on both sides, then arrange on a sheet pan.
Add the plums to another sheet pan, season with salt and pepper, drizzle with honey, add the thyme sprigs, drizzle with a little olive oil, then use your hands to toss until coated.
Place the bread and the plums in the oven and bake until the bread is lightly toasted, about 10 minutes, and until the plums are caramelized, about 20 minutes. Remove the thyme sprigs from the plums, and while still warm, transfer them to a bowl.
Arrange the toasted bread on a platter, then drape a piece of prosciutto over each one. Spoon the plums on each, then use your hands to break apart pieces of the burrata and lay over top. Sprinkle with some fresh thyme leaves and drizzle with more olive oil and a sprinkle of flaky sea salt, if desired. Serve immediately.