1. Adjust oven rack to upper-middle position and heat to 400F/200C. Combine butter, pimentos, and sage in a small bowl and stir with a fork until combined.

2. Pat chicken dry with paper towels and season with salt and pepper. Spread butter mixture on top of each chicken breast. Working with 1 breast at a time, lay 2 slices of prosciutto on counter, slightly overlapped. Lay chicken, buttered side down, in center of slices, fold prosciutto up and around chicken, then press ends of prosciutto together gently to adhere.  Repeat with remaining 2 slices prosciutto and chicken breast.

3. Heat oil in 12" skillet over medium-high heat until just smoking. Add chicken, seam side down, and cook until well-browned, about 2 minutes per side. Transfer chicken to baking sheet and bake until chicken registers 160F, about 20 minutes. Turn off skillet heat for now.

4. About 10 minutes into chicken bake time, turn skillet heat back on, to medium.  Add green beans and water; cook, covered, until beans are bright green and tender, 8-10 minutes. Uncover and cook until water has evaporated, about 1 minute. Add minced garlic and combine; cook until fragrant, about 30 seconds. Season with salt and pepper to taste and serve with chicken.