Cut the potatoes in half, place them in a large saucepan, add enough water to cover and the salt. Cook over high heat for about 20 to 25 minutes, or until they are soft enough for a knife to pierce easily. Drain and set aside until cool enough to handle.
While the potatoes are cooking, cut the broccoli into individual bite-sized florets. Trim and peel the stems and cut crosswise into ½-inch rounds. You will have about 3 cups small broccoli florets and stems. Steam, blanch, or microwave the broccoli with a tablespoon of water until it is bright green and still a bit firm. Drain and set aside.
In a medium-sized bowl, make the dressing: whisk together the vinegar, honey, sugar, mustard, celery seeds, white pepper and mayonnaise. Peel the potatoes and cut into 1-inch cubes. Place the still-warm pieces into the dressing and stir gently to coat. Add the broccoli, also still warm, and stir gently to coat. Add the carrots and stir gently before serving. Serve immediately or refrigerate and serve cold.