Drain the mozzarella and cut into cubes approximately 0.5 cm on each side.
For the marinade, coarsely crush the pink pepper and press the garlic. Mix the pepper, garlic, vinegar, oil, thyme, and oregano together, then pour the mixture over the mozzarella cubes. Let the mozzarella soak for 15 minutes, gently turning it several times.
While the mozzarella marinates, peel the cucumber, halve it, and remove the seeds. Cut the flesh into cubes roughly the same size as the mozzarella. Clean the radicchio, place the outer leaves on 4 serving plates as a bed, and cut the remaining inner leaves into fine strips.
Combine the marinated mozzarella, cucumber cubes, and radicchio strips. Distribute the mixture on the plates over the radicchio leaf bases.
Serve with a glass of Prosecco or a light white wine.