Combine all of the pesto ingredients except the oil in a blender, food processor, or immersion blender. Process until mostly smooth. With the motor running, slowly add the oil until you reach a smooth and creamy consistency. Toss the chicken in the pesto and marinate in the fridge for 4-12 hours.
Preheat the oven to 375°F.
Cook the wheat berries according to the packet instructions, drain and cool slightly.
In a medium saucepan, bring the lentils, water, bay leaf, and a pinch of salt salt to a boil. Reduce heat to a simmer and cook for 25-30 minutes until the lentils are tender. Drain.
In a large bowl, toss together the wheat berries, lentils, cucumber, garlic, parsley, olive oil, lemon juice, tomatoes, salt and pepper.
Remove the chicken from the marinade, discard marinade and season chicken with salt and pepper. Roast on a baking sheet until cooked through, about 22 minutes. Rest for 5-10 minutes before slicing.
Bring a large pot of salted water to a boil for the green beans. Prepare an ice bath by filling a large bowl with water and ice. When the water boils, add the green beans for 2-4 minutes, until bright green and crisp-tender, then immediately put them into the ice bath to stop cooking. Take them out of the ice bath after 1-2 minutes.
Add all of the caper dressing ingredients except the oil to a blender. Purée until smooth. With the motor running, drizzle in the oil until emulsified. Season to taste.
Rinse out the blender and add all of the aioli ingredients except the oil. Purée until smooth. With the motor running, drizzle in the oil until emulsified. Season to taste.
To serve, divide the wheat berry salad between bowls. Top with cherry tomatoes, radishes, chicken breasts and green beans and serve with caper dressing and aioli on the side.