In a small jar, combine the olive oil, white balsamic vinegar, fresh thyme, honey, salt, and black pepper. Shake vigorously to emulsify the herb vinaigrette and set aside.
Preheat an outdoor grill or indoor cast-iron grill pan to medium-high heat. Lightly brush the peach wedges and halloumi slices with a small amount of olive oil.
Place the halloumi slices on the hot grill. Cook for 2 to 3 minutes per side until deep, golden grill marks appear. Transfer to a plate.
Place the peach wedges on the grill. Cook for 1 to 2 minutes per side until slightly softened and charred. Remove from the heat.
In a large serving bowl, toss the baby arugula, sliced cucumber, and red onion with half of the herb vinaigrette.
Arrange the warm grilled halloumi and peaches on top of the greens. Drizzle with the remaining vinaigrette and serve immediately.