Caramelize onions: Heat a large nonstick skillet on medium-high and lightly mist with cooking spray. Add onions and cook, stirring frequently, until fragrant, 1 to 2 minutes. (NOTE: Skillet will be quite full to start, but onions shrink as they cook.) Reduce heat to medium-low and cook, stirring occasionally, until very soft, sticky and light caramel in color, about 1 hour. Remove from heat, transfer to a large plate and let cool at room temperature for 10 minutes.
Meanwhile, preheat oven to 425°F. In a medium bowl, combine Panko, thyme, salt and pepper. In a separate medium bowl, lightly beat egg whites; set aside. With your hands, gently press goat cheese into 4 1-oz balls and flatten each into 1/2-inch-thick patties. Dip patties in egg, turning to coat, then gently press both sides into Panko mixture; repeat process to coat each patty twice. Transfer to a small tray, cover loosely with plastic wrap and freeze for 20 minutes.
Meanwhile, prepare dressing: In a small bowl, whisk lemon juice, Dijon and additional salt and pepper; set aside.
Line a medium baking sheet with foil and mist with cooking spray. Arrange cheese patties in a single layer on 1 half of sheet and mist tops with cooking spray. Bake for 6 minutes. Flip patties and bake for 6 more minutes, until golden. Turn oven heat off and remove sheet from oven. Add pecans to remaining half of sheet. Return to oven until pecans are warm, about 3 minutes.
Meanwhile, in a large bowl, combine arugula, strawberries and dressing. Toss gently to coat and divide among serving plates. Top with onions, pecans and cheese patties, dividing evenly.