Make the balsamic drizzle: combine balsamic vinegar and honey in a small saucepan. Simmer over medium heat for 8 to 10 minutes until reduced by half and slightly syrupy. Set aside to cool.
Dice tomatoes. Transfer to a colander, sprinkle with salt, toss, and let drain for 10 minutes. Press gently to remove excess liquid. Discard the liquid.
Combine drained tomatoes, minced garlic, olive oil, balsamic vinegar, and black pepper in a bowl. Stir and let sit for 15 minutes. Do not add basil yet.
Preheat oven to 400°F. Arrange baguette slices on a baking sheet. Brush both sides with olive oil. Toast 8 to 10 minutes, flipping once, until golden. While hot, rub each slice firmly with a peeled garlic clove.
Right before serving, fold fresh basil into the tomato mixture. Taste and adjust salt.
Spoon tomato topping generously onto each toast. Drizzle balsamic reduction over the top. Finish with flaky salt and extra basil leaves. Serve immediately.