In a medium bowl, whisk together 1/4 cup olive oil, the vinegar, mustard, anchovy paste, garlic and 1/4 teaspoon each salt and pepper; transfer half of the vinaigrette to a small bowl, discarding any large pieces of garlic. Add the chicken to the remaining vinaigrette and toss to coat; cover and refrigerate for 30 minutes or up to overnight.
Preheat a foil-lined grill to medium. Brush the corn with 1 tablespoon olive oil. Thread the chicken onto four 10-inch skewers and discard the marinade. Place the chicken kebabs and the corn on the grill, cover and cook, turning occasionally, until lightly charred and the chicken is firm, 10 to 15 minutes. Brush the corn with the remaining 1 tablespoon olive oil; season with salt and pepper. Sprinkle the corn and chicken with 3 tablespoons parmesan cheese.
In a bowl, toss the romaine with the remaining vinaigrette and 2 tablespoons cheese. Serve the kebabs with the salad and corn.