Hull the strawberries and then trim the stem ends so they will sit flat in the pan. In a medium bowl combine the berries with the sugar, vanilla, and liqueur (or lemon water) and toss to coat.
Prepare the grill for cooking over high heat (500°F).
Use the softened butter to generously coat the bottom and sides of an 8-by-8-inch metal baking pan. The pan should be just large enough to hold the berries in a single layer with their sides almost touching (this allows the berries to gently support one another as they begin to soften). Remove the berries from the bowl and line them up so that they fit snuggly, pointing up, in the prepared pan. Pour the contents from the bowl over the berries.
Place the pan over high heat, close the lid, and cook until the strawberries are bubbling and beginning to slump, 8 to 12 minutes. Cooking times will vary depending on the variety, size, and ripeness of the strawberries. Watch closely to catch them before they collapse. Remove from the grill.
Spoon the pan juices over the berries to moisten them and let cool for 5 minutes. Carefully cut them into quarters or leave whole. Ladle berries over ice cream.