Directions:
Make vinaigrette:
Whisk together all vinaigrette ingredients, except oil and chives, with 1/2 teaspoon salt and 1/4 teaspoon pepper. Add oil in a slow stream, whisking until emulsified. Whisk in chives.
Make radish-cucumber salad:
Boil water with salt, sugar, garlic, and peppercorns in a 4-quart pot, uncovered, 10 minutes.
While brine boils, trim and halve radishes. Halve cucumbers lengthwise and cut crosswise into 1/2-inch-thick slices.
Remove brine from heat. Add radishes and cucumbers and let stand, uncovered, 10 minutes. Drain in a colander, discarding garlic and peppercorns. Transfer radishes and cucumbers to an ice bath to stop cooking, then drain well in colander.
Transfer to a large bowl and chill, uncovered, about 20 minutes.
Make chickpea salad:
Stir together chickpeas, onion, 1/4 cup vinaigrette, and salt and pepper to taste.
Cook green beans:
Cook green beans in a large pot of well-salted boiling water, uncovered, stirring occasionally, until just tender, 3 to 6 minutes. Drain. Transfer to a large ice bath to stop cooking. Drain again and pat dry.
Grill mushrooms and chicken:
Prepare grill for direct-heat cooking over hot charcoal (medium-high heat for gas); see Grilling Procedure .
Toss mushrooms with 2 tablespoons vinaigrette and marinate 10 minutes.
Grill mushrooms in 2 batches on oiled grill sheet, covered only if using a gas grill, stirring frequently, until golden-brown, about 5 minutes per batch. Toss hot mushrooms with 2 tablespoons vinaigrette.
Season chicken with 1/2 teaspoon each of salt and pepper. Oil grill rack, then grill chicken over medium-hot charcoal (medium heat for gas), covered only if using a gas grill, turning chicken occasionally and moving it as necessary to avoid flare-ups, until just cooked through, 8 to 10 minutes total. Transfer to a cutting board and let rest 10 minutes.
Cut into 1/2-inch-thick slices and toss with pesto in a large bowl.
Dress salads and assemble dish:
Toss brined cucumbers and radishes with parsley, 3 tablespoons vinaigrette, and salt and pepper to taste.
Stir mint into chickpea salad.
Toss beans with 2 tablespoons vinaigrette and salt and pepper to taste. Sprinkle with almonds.
Toss tomatoes with 3 tablespoons vinaigrette, basil, and salt and pepper to taste.
Toss lettuce with 1 tablespoon vinaigrette.
Arrange chicken, mushrooms, and salads side by side on a large platter and serve remaining vinaigrette on the side.
Cooks' notes:
Vinaigrette, without chives, can be made 1 day ahead and chilled. Add chives just before serving.
Radish-cucumber salad, without parsley, can be made 1 day ahead and chilled. Add parsley just before serving.
Chickpea salad, without mint, can be made 1 day ahead and chilled. Add mint just before serving.
Haricots verts can be cooked 1 day ahead and chilled in a sealable bag lined with paper towels.
Mushrooms and chicken can be cooked in batches in an oiled hot 2-burner grill pan over medium-high heat.