In bowl, place onion slices, lemon juice, sugar, and salt. With your hands, massage onions until tender and pink, about 2 minutes. Chill until ready to use.
Heat grill to medium-high. Brush both sides of mushroom caps with 2 tsp (10 mL) oil and season with paprika. Place mushrooms on grill grate and heat until tender with grill marks, about 2 minutes per side. When cool enough to handle, cut mushrooms into 1/2 in (1.25 cm) strips.
In skillet over medium, heat remaining oil; add tomatoes and garlic, and heat until tomatoes have softened and begun releasing their juices, about 3 minutes. Stir in red wine vinegar and season with salt and pepper.
To make dressing, whisk together pesto, olive oil, and vinegar.
To assemble salad, divide greens among serving plates and top with chickpeas, mushrooms, tomatoes, artichokes, bocconcini, onion, and olives. Drizzle on pesto dressing.