Whisk together the shallot, vinegar, mustard, garlic paste, and a large pinch of pepper in a large bowl. Add 2 tablespoons of the olive oil in a slow steady stream, whisking constantly, until the dressing emulsifies. Add the tomatoes to the vinaigrette and set aside until ready to serve, up to 2 hours.
Mix 1/4 teaspoon each salt and pepper in a small cup. Pound the chicken to 3/8 inch thick between two sheets of waxed paper or plastic wrap. Season with some of the salt and pepper.
Beat the egg in a wide shallow bowl until smooth. Spread the breadcrumbs on a plate. Dip a chicken breast into the egg, coating thoroughly and shaking off any excess. Then dip it into the breadcrumbs, patting on the crumbs so both sides are well coated and shaking off any excess. Lay the breaded breast on a sheet of waxed paper. Repeat with the other chicken breast.
Heat the butter and 2 teaspoons olive oil in a heavy, large (preferably 12-inch) skillet over medium-high heat. When the oil is hot enough to sizzle a breadcrumb, add the chicken breasts, keeping them separated. Cook, shaking the pan occasionally, until golden brown, about 2 minutes. Season each breast with some of the salt and pepper, gently turn over with tongs, taking care to keep the golden crust intact, and cook until the second side is golden brown, 1 to 2 minutes. Watch carefully and don't let the crumbs get too dark. Season with salt and pepper and drain for just a moment on paper towels.
To serve, add the arugula to the vinaigrette and tomatoes and toss. Arrange the sautéed chicken on warm serving plates and top each with a tall pile of the salad.