Place potatoes, carrot, garlic, bay leaf, and 1 tablespoon salt in a medium saucepan; fill with cold water to cover by 1 inch. Bring to a boil over high. Reduce heat to medium, and simmer, uncovered, until potatoes are fork-tender, 8 to 10 minutes, adding corn and peas during final 3 minutes of cooking time. Drain potato mixture; discard bay leaf and garlic cloves.
Transfer warm potato mixture to a medium bowl. Add vinegar and remaining 3/4 teaspoon salt; toss to combine. Stir in Kewpie mayonnaise and black pepper. For a creamier salad, use a fork to mash a portion of the potatoes. Chill potato salad in refrigerator, uncovered, until cooled to room temperature, about 20 minutes, stirring occasionally.
Stir cucumber and scallions into potato salad; season to taste with salt, and garnish with black pepper. Serve potato salad immediately, or spread on sliced bread for a creamy sandwich, if desired.
