Sprinkle the garlic and sage on the chicken cutlets; season with salt and pepper. Wrap 2 or 3 slices prosciutto around each cutlet.
Heat a large nonstick skillet over medium heat; add 1 tablespoon olive oil. Working in batches if necessary, add the chicken and cook, flipping once, until the prosciutto is golden brown and slightly crisp, 6 to 8 minutes. Transfer to a plate.
Meanwhile, whisk half of the lemon juice, the mustard, a pinch of salt and a few grinds of pepper in a large bowl. Gradually whisk in the remaining 2 tablespoons olive oil until smooth. Add the lettuce and tomatoes; toss to coat.
Add the chicken broth and the remaining lemon juice to the skillet; cook over medium heat until slightly reduced, 1 to 2 minutes. Add the butter and cook, stirring with a rubber spatula, until the butter melts and the sauce thickens, 1 to 2 minutes. Serve the chicken with the salad. Drizzle with the pan sauce.