Combine vinegar, onion, and salt in a large bowl; let stand 10 minutes. Add 1/3 cup oil, the mustard, honey, thyme, and pepper; whisk until combined.
Add chicken and celery; toss. Gently fold in chopped tomato and 1/4 cup basil. Spoon mixture onto toasts. Top with remaining basil, pepper to taste, a drizzle of olive oil, and a sprinkling of flaky salt. Serve with sliced tomatoes.