TO MAKE THE SALAD: Toss the celery, cucumber, bell pepper, herbs and greens into a bowl. Pour the oil, lemon juice and vinegar into a small jar, season with salt and pepper and shake to blend. Set the salad and vinaigrette aside until needed. (If you’re going to chill the chicken breasts, cover and refrigerate the salad.)
WORKING AHEAD: If you can bread the cutlets and give them a few hours in the fridge before sautéing them, do it. A chill gives the coating time to firm and dry a bit, so you get a crisper cutlet.
TO MAKE THE CHICKEN: Center a rack in the oven and preheat it to 300°F, to keep the first batch of cutlets warm while you cook the rest. Line a baking sheet with parchment paper or a double layer of paper towels. Have another baking sheet lined with parchment or a rack to hold the breaded cutlets.
Sandwich the cutlets between sheets of parchment or wax paper and pound them with the bottom of a skillet or the at side of a meat tenderizer.
Set out three shallow bowls (I use soup plates). Put about 1⁄2 cup of breadcrumbs in each of two bowls and crack the eggs into the third bowl; put the egg bowl between the other two. Season the crumbs and eggs with salt and pepper and lightly beat the eggs to break them up.
One by one, dredge the cutlets in the first bowl of crumbs, run them through the eggs and then coat them in the second bowl of crumbs, placing the breaded cutlets on the lined baking sheet or the rack. Replenish the bread crumbs as you go, if needed. If you have time, chill the cutlets, uncovered. (The breaded cutlets can be refrigerated for up to 8 hours.)
Set a large skillet—nonstick is great here—over medium heat. Add 1 tablespoon of the butter and 2 tablespoons of the oil, and when the butter is melted and the bubbling has subsided, slip in 2 cutlets, or as many as fit comfortably in the pan. Cook until the breading is golden on the underside—adjust the heat and tilt the pan as needed so that the chicken, not the butter, browns—then carefully turn the cutlets over to cook and brown the other side. You’ll need about 3 minutes on each side, but this will vary according to thickness—it’s best to cut into the center of a cutlet to test. When the cutlets are golden and cooked through, transfer them to the second lined baking sheet and pop them into the oven to keep warm.