In a mixing bowl, combine the diced tomatoes, red onion, olive oil, vinegar, honey, basil, salt, and pepper. Stir well and let it sit at room temperature for 15 minutes while you prep the burgers.
Slice each halloumi block in half lengthwise to make 4 slabs, about 1 cm / 0.4 inch thick. Pat both sides dry with a paper towel.
Brush both sides of the halloumi slabs with the 1 tablespoon of olive oil.
Heat the grill pan over medium-high, about 200 C / 400 F, for 2 minutes until it's hot.
Lay the halloumi slabs on the pan and don't move them for 2 to 3 minutes, until deep golden grill marks form. Flip and grill the other side for 2 minutes, until golden and slightly softened but still holding its shape.
Transfer the halloumi to a plate. Butter the cut sides of the buns and toast them cut-side down in the same pan for 1 to 2 minutes, until golden.
In a small bowl, mix the grated garlic into the mayonnaise.
Spread the garlic mayo on the bottom half of each bun. Top with arugula, a halloumi slab, and a spoonful of tomato relish, then close with the top bun.
Serve right away while the halloumi is still warm and soft.