Halve or quarter the cherry tomatoes and place in a salad bowl. Finely chop the shallot and add to the bowl, along with the vinegar, 2 tablespoons olive oil and ¼ teaspoon each salt and pepper. Toss gently to combine.
Coarsely shred the Havarti. Lightly toast the bread then arrange on a baking sheet. Rub the garlic cloves over one side of the toasts and drizzle with the remaining 2 tablespoons olive oil. Divide the cheese evenly among the toasts and sprinkle the oregano over top.
Broil the toasts until the cheese is melted and bubbling. Remove from the broiler and transfer each toast to a plate. Top with the tomato salad and garnish with the chopped chives. Serve at once.