Remove stem ends and any strings from beans. In a 6- to 8-quart pan over high heat, bring 2 quarts water to a boil. Add beans and cook until tender-crisp to bite, about 4 minutes. With a slotted spoon, transfer beans to a bowl of ice water to cover. When cool, lift beans out and put in a wide, shallow bowl.
Add onions to boiling water and cook just until tender when pierced, about 4 minutes. With slotted spoon, transfer to ice water.
Meanwhile, scrub potatoes and yams. Put in boiling water, adding more water if needed to cover. Cook until vegetables are tender when pierced, about 25 minutes. Drain.
While potatoes and yams cook, drain onions and peel.
In a bowl, whisk together garlic, mustard, vinegar, and oil.
Peel yams; cut potatoes and yams into 1/2-inch slices.
Place potato and yam slices on a barbecue grill over very hot coals or very high heat on a gas grill (you can hold your hand at grill level only 1 to 2 seconds); close lid on gas grill. Cook, turning once, until grill marks develop on both sides, 2 to 4 minutes total.
Add potatoes, yams, onions, tomatoes, and garlic dressing to beans. Mix gently; season to taste with salt and pepper.
Arrange arugula on a platter, spoon salad onto leaves, and sprinkle with mint.