Make the marinade / dressing:

Add garlic clove (or chopped green garlic), scallions, and herbs to the bowl of a food processor or blender. Pulse about 10 to 12 times, until herbs are finely minced. Add anchovies, yogurt, mayonnaise, lemon juice and zest and process until smooth, about 1 minute, scraping down bowl as needed. Taste for seasoning and add salt and pepper to taste. If dressing is too thick, add a couple of tablespoons water. You should be able to drizzle it off a spoon. Set aside. Note: dressing can be made up to 3 days ahead.

Cut chicken breasts in half lengthwise, making two thinner cutlets. Place each cutlet between 2 pieces of plastic wrap and pound until even and about 1/4-inch thick. Add pounded chicken to a sealable plastic bag and pour about a cup of the marinade into bag. Move the chicken around, making sure every piece is coated, then seal the bag and refrigerate for at least 2 hours, and up to 24 hours. Transfer the rest of the dressing to a sealable jar and refrigerate.

Prepare a grill for high heat and take the chicken out of the refrigerator. Brush onions with olive oil and season with salt and pepper. When grill is hot, clean and oil grill grates. Grill chicken and onions until cooked through and brown in spots, about 4 to 5 minutes each side, depending on how hot your grill gets. Transfer chicken and onions to a clean platter and drizzle some of the reserved dressing over. Serve warm or room temperature.