For the dressing: Smash the garlic cloves, sprinkle with the salt, and, with the flat side of a large knife, mash and smear the mixture to a coarse paste. Put in a bowl and whisk with the mayonnaise, vinegar, and black pepper to taste.
For the salad: Put the water, vermouth, garlic, salt, thyme, bay leaf, peppercorns, and potatoes in a large saucepan and bring to a boil. Add the carrots, lower the heat, and cook until the vegetables are tender but not mushy, about 5 minutes. Stir in the radishes, then immediately drain all the vegetables in a colander in the sink. Remove and discard the garlic, thyme, bay leaf, and peppercorns. Cool slightly and toss the vegetables with the dressing. Cover and refrigerate about 30 minutes. (The salad can be prepared up to this point a day ahead.)
About 10 minutes before serving, toss the cucumber, tomatoes, and scallions in a small bowl with salt and black pepper to taste. When ready to serve, fold the cucumber mixture and herbs into the potato salad. Serve with lemon wedges.
If you have a mandoline, use it to make slicing the vegetables for this salad a breeze.