Preheat the oven to 350°F. Line a baking sheet with parchment paper or aluminum foil.
Lightly brush each of the baguette slices with the olive oil and place them side by side on the baking sheet (they can be touching, but not overlapping – you may need two baking sheets, or do this in two batches). Sprinkle the bread with coarse salt.
Bake for 4 to 7 minutes until lightly brown. Remember that they will harden as they cool, so take them out before they get too crisp. If desired, lightly rub the flat side of a cut garlic half against the top of the bread.
Combine the tomatoes, olive oil, basil, garlic, salt and pepper in a medium-sized bowl. You can let this, and the crostini, sit at room temperature for up to an hour before topping the crostini.
Lay the crostini slices on a serving tray and top each with a spoonful of crostini, or simply serve the tomato mixture in a bowl with the crostini on the side. When you scoop the tomatoes onto the bread, use a slotted spoon, or tilt the spoon a bit, to allow some of the liquid to drain off…or use your clean hands. If you are topping the crostini, do it right before serving or the bread will become soggy.