Combine the oil, vinegar, basil, chives, shallots, mustard, garlic, salt, and pepper in a container with a tightly sealed top (a small canning jar works well). Cover tightly and shake well, until well blended. Use immediately or refrigerate, covered, until ready to use. Shake well before serving.
In a large bowl, toss the watercress, red tomatoes, yellow tomato, onion, cucumber, bell pepper, radishes, mushrooms, and hearts of palm. Drizzle with the vinaigrette and toss to coat.