Place brussels sprouts on a rimmed baking sheet. Drizzle with 2 tablespoons olive oil and season generously with salt and pepper. Toss to coat and arrange brussels sprouts on baking sheet, cut side down. Transfer to oven and roast until brussels sprouts are deeply golden, about 20 minutes, flipping brussels sprouts to cut-side up three-fourths of the way through cooking.
Meanwhile, in a medium pot, combine quinoa, a large pinch of salt and 2 cups water. Bring to a boil over high heat. Top with a lid and reduce heat to low. Cook 12 minutes. Remove lid and continue to cook, stirring occasionally and adjusting heat if necessary, until water is evaporated and quinoa is tender but still al dente, about 4 minutes more.
Preheat a skillet over medium-high heat. Add remaining 2 tablespoons olive oil and swirl to coat. Add breadcrumbs, season with salt and pepper, and toast, stirring occasionally, until golden and crisp, about 3 minutes. Remove from heat and sprinkle with zaatar.
Place greens, chicken and tomatoes in a large bowl and drizzle with enough pesto vinaigrette to coat. Divide among serving bowls. Top with quinoa, fresh ricotta, brussels sprouts and toasted breadcrumbs. Pass additional dressing at the table.
For the Pesto Vinaigrette:
Combine pesto, olive oil and balsamic vinegar in a small bowl. Season to taste with salt and pepper.