How to Make It
In a large bowl, whisk 1 tablespoon each of both vinegars with the mustard. Whisk in 2 tablespoons of the olive oil and season with salt and pepper. Add the greens and chopped eggs but don't toss.
In a large skillet, melt 2 tablespoons of the butter in the remaining 2 tablespoons of oil. Cook over moderately high heat until the butter begins to brown, about 1 minute. Add the prosciutto and cook, stirring occasionally, until crisp, 2 to 3 minutes. Add the remaining 1/4 cup of balsamic vinegar and 2 tablespoons of red wine vinegar and cook until slightly reduced, about 2 minutes. Remove from the heat. Swirl in the remaining 3 tablespoons of butter until melted.
Toss the greens, season with salt and pepper and transfer to plates. Spoon the hot dressing over the greens and serve.