Preheat the oven to 350F. Lightly grease a large casserole dish (3.5L) or spray with cooking oil.
Bring the quinoa and water to a boil in a medium saucepan. Cover, reduce to a simmer and cook for 10 minutes. Turn the heat off and leave the covered saucepan on the burner for another 6 minutes. Fluff with a fork and cool. Evenly spread the cooked quinoa over the bottom of the casserole. Set aside.
Cook the broccoli in a steamer or medium saucepan until just tender or al dente. Place the broccoli in a medium bowl. Add the peas to the broccoli.
Heat the oil in a large nonstick pan over medium-high heat. Brown the chicken until cooked and the juices run clear, about 5 minutes on each side. Remove chicken from the pan, leaving the juices/oil still in the pan. Add the shallot and sauté until clear. Add the chicken back in and just toss until mixed.
Add the chicken to the broccoli and peas.
Combine the Greek yogurt, mayonnaise, milk, lemon juice, mustard, curry powder and basil in a medium bowl and mix well. Add more milk if you want the sauce a bit thinner. Add the sauce to the broccoli, peas and chicken mixture. Stir until mixed well. Spread this mixture over the quinoa. Sprinkle the cheese on top.
Cover with aluminum foil and bake on the center oven rack for 15 minutes. Remove the foil and bake another 10 minutes, or until the cheese is melted and bubbly.