Rub canola oil on both sides of the chicken breasts and place each breast in a separate sandwich-size plastic bag. Use a mallet or heavy skillet to pound each chicken breast until it's about ½ inch thick and has roughly doubled in size.

Press 3 sage leaves onto the skinned (smooth) side of each chicken breast. Wrap one slice prosciutto around each chicken breast, leaving about 1½ inches exposed on both ends of the breast. Season with kosher salt and freshly ground black pepper on both sides. Repeat with the remaining prosciutto and chicken.

Heat ¼ cup of the canola oil in a large skillet over high heat for 2 to 3 minutes, until the oil's almost smoking (you will begin to smell the oil at that point). Place two of the chicken breasts in the skillet, sage side down, and sear them for 2 minutes. Flip the chicken and cook for 2 minutes more. Transfer the chicken breasts to a plate, wipe out the pan and repeat, heating the remaining ¼ cup of canola oil and cooking the other two chicken breasts in the same way.

Heap the greens onto four plates, dividing them evenly, and sprinkle them with sea salt. Lay the chicken breasts first-cooked side up on top of the greens. Pour the juices from the plate the chicken was resting on into the pan. Add the sage pesto to the pan and warm it over medium-high heat, stirring constantly, until it bubbles. Add the vegetable broth and cook until the sauce is reduced by a third and thickens slightly. Pour the sauce over the chicken breasts and drizzle with the lemon-infused olive oil.