Add the potatoes to a large pot and cover them with cold water, ensuring it extends about an inch above the potatoes.
Add the crushed garlic cloves, lemon half, and salt.
Bring to a boil over medium-high heat, then reduce to a simmer and cook for 15-20 minutes, or until the potatoes are fork tender.
Drain the potatoes and spread them onto a sheet pan or large platter. While they’re still warm, drizzle them with 1 tablespoon red wine vinegar and gently toss.
Allow the potatoes to cool slightly.
In a medium bowl, whisk together the red wine vinegar, lemon juice, Dijon mustard, garlic, oregano, salt, and black pepper.
Slowly stream in the olive oil while whisking continuously until emulsified.Taste and adjust seasoning as needed. If desired, add Calabrian chili flakes, Aleppo pepper, or honey.
Thinly slice the red onion and rinse under cold water to mellow its sharpness.
Chop the parsley, tear the basil, halve the cherry tomatoes and mozzarella, slice the roasted red peppers, and drain the capers.
Add the potatoes, red onion, parsley, roasted red peppers, olives, capers, cherry tomatoes, mozzarella, basil, and dried oregano to a large mixing bowl. Stir the chopped parsley and basil into the vinaigrette, then pour approximately 3/4 cup of the dressing over the salad. Gently fold until evenly coated.
Let the salad rest for 15-20 minutes before serving to allow the flavors to meld. Taste and adjust with additional dressing, salt, pepper, or lemon juice as needed.
Serve slightly warm or chilled. Garnish with additional basil, parsley, or a few extra capers if desired. Enjoy!