Heat 15 mL (1 tbsp.) olive oil in a large skillet over medium heat, and fry onions and garlic without letting them brown. Add cumin and stir mixture well for about 1 minute. Transfer to food processor bowl.
Add chickpeas, beet, breadcrumbs, eggs, and tahini. Season with salt and pepper to taste. Process until coarsely puréed.
With your hands, shape mixture into 20 small balls and place on a baking sheet lined with parchment paper. Brush falafels with remaining olive oil. Bake in the oven for about 30 minutes, or until crispy on the outside and hot in the middle.
In the meantime, prepare the sauce: combine yogurt, lemon zest and juice, and half of parsley and mint in a small bowl.
Spoon yogurt sauce into each pita and stuff with falafels, radish, cucumber, and remaining parsley and mint.