Put the strawberries in a bowl, sprinkle with 1 to 2 tablespoons of the sugar (depending on their sweetness), and toss gently to coat. Add the vinegar and toss again. Let macerate at room temperature until you’re ready for dessert, up to several hours.
Beat the eggs lightly in a large shallow bowl, then beat in the milk, the remaining 1 tablespoon sugar, vanilla, and salt. Refrigerate until you’re ready to use it (or up to several hours).
If using a gas grill, heat it for medium direct cooking. If using a charcoal grill, after taking dinner of, add more coals if necessary and close the lid until you’re ready to cook. In either case, clean the grates.
Give each slice of bread a quick dip in the egg wash on both sides, put on a large plate, and immediately take out them to the grill. Put the bread on the grill directly over the fire. Close the lid and cook until the bread develops grill marks, 2 to 3 minutes per side. Transfer to a clean platter. To serve, top each grilled toast with a spoonful of the balsamic strawberries. Let diners add freshly ground black pepper to their strawberries, if they like.