The day before serving, heat the olive oil over medium-high heat in a large skillet. Add the onions and quickly sauté just until they begin to soften, about 4 minutes. Stir in 2 tbls. of the vinegar, stirring to coat. Transfer to a bowl, cover and refrigerate.
Two hours before serving, remove the onions from refrigeration and stir in the remaining vinegar.
Roll each slice of prosciutto loosely and arrange at one end of a large platter; pile the strawberries in the center and the onions at the other end. Tuck the slices of cheese around the platter and serve.