Make the dressed cherries: Add the halved cherries to a bowl with the sherry vinegar, olive oil, basil, flaky sea salt, and cracked black pepper. Gently toss and let sit for 10–15 minutes so the cherries lightly marinate and release some of their juices.
Toast the bread: Lightly brush both sides of the bread with olive oil and toast in a skillet, grill pan, or under the broiler until golden and crisp around the edges.
Add the Gorgonzola: While the bread is still slightly warm, spread each toast generously with softened Gorgonzola.
Add the ham: Loosely layer the thinly sliced ham over the Gorgonzola, folding it rather than laying it flat to give the toast some height.
Add the cherries: Spoon the basil-dressed cherries over each toast, along with a little of the dressing from the bottom of the bowl.
Finish and serve: Top with fresh basil, flaky sea salt, and cracked black pepper and serve immediately.