Make salad: Heat oil with chile and salt in a small saucepan over medium heat until fragrant, 2 minutes. Add quinoa and toast, stirring often, until it starts to turn golden in spots, 4 to 6 minutes. Stir in 2/3 cup water, cover, and simmer 15 minutes. Turn off heat and let sit 5 minutes, then pour quinoa onto a plate and let cool. Discard chile.
Meanwhile, put remaining salad ingredients except for cheese in a wide bowl and set aside.
Make dressing: In a small bowl, combine 2 tsp. water with all dressing ingredients except oils. Set aside 10 minutes; whisk in oils.
Gently toss quinoa with 2 tbsp. dressing. Spoon onto 4 dinner plates. Mound salad mixture on top and tuck in cheese. Season generously with more pepper and serve with remaining dressing to add to salads.