Cook each kind of potato separately. Start potatoes in cold, salted water, bring to a boil, and cook until tender, yet not falling apart. Drain and cool on parchment paper. Cut the tiny red potatoes into wedges. Toss in olive oil. Place on hot grill and grill on both sides until grill marks show and they are warm throughout. Combine all ingredients for the Potato Dressing allow to steep for about 1/2 hour. Toss the potatoes in the steeped dressing. Set aside. Blend the yellow vine ripe tomatoes in a blender with extra virgin olive oil, vinegar, garlic, and salt until smooth. Strain through a fine sieve. Place all the miniature tomatoes in a bowl, sprinkle with salt and basil, cover with yellow tomato vinaigrette and gently mix. Adjust seasoning with more vinegar, salt, and oil, if needed. Arrange grilled dressed potatoes in the center of each of 4 plates and surround with the tomato salad. Top each salad with 2 ounces goat cheese. Drizzle with basil oil and top with a basil sprig.