Prepare the dressing: Smash the garlic cloves, sprinkle with kosher salt, and use the flat side of a large knife to mash into a coarse paste. Whisk this with mayonnaise, white wine vinegar, and freshly ground black pepper. Set aside.
Cook the potatoes: In a large saucepan, combine water, vermouth, smashed garlic, salt, thyme, bay leaf, and peppercorns. Bring to a boil and add the sliced potatoes.
Blanche the vegetables: Add carrots to the boiling pot, reduce heat, and cook for about 5 minutes until just tender. Stir in radishes, then drain all vegetables in a colander immediately.
Cool and toss: Remove and discard the bay leaf, garlic cloves, thyme, and peppercorns. Toss the warm vegetables with the prepared dressing to coat evenly.
Chill: Cover the salad and refrigerate for at least 30 minutes to allow the flavors to meld. (You can make it up to a day ahead.)
Prepare fresh vegetables: Just before serving, toss cucumbers, tomatoes, and scallions with a light sprinkle of salt and pepper.
Assemble and serve: Fold in the cucumber mixture and fresh herbs into the potato salad. Serve chilled with lemon wedges for a burst of freshness.