Preheat oven to 425°F (220°C).
On a baking sheet, toss zucchini, summer squash, and 1 garlic clove with olive oil, salt, and pepper. Roast for 45-50 minutes until tender and most moisture has cooked out, turning halfway.
Let squash cool slightly. Transfer roasted squash and the single garlic clove to a food processor. Add smoked paprika, cayenne pepper, cooked quinoa, panko bread crumbs, and fresh basil. Pulse until combined.
Line a baking sheet with parchment paper. Form the mixture into 10-12 patties (they will be sticky). Chill patties in the refrigerator for at least 1 hour, or up to 1 day.
Heat 1 tbsp olive oil in a large skillet over medium-high heat. Cook 2-3 patties at a time until golden brown, about 5-8 minutes per side. Be gentle when flipping.
For the Whipped Feta: Combine feta cheese, Greek yogurt, olive oil, and the 3 reserved roasted garlic cloves in a food processor.
Puree until smooth and creamy (about 5 minutes), scraping down sides.
Add pepper, dill, and lemon zest; pulse to combine. Thin with milk if needed.
Serve the zucchini burgers on warmed buns, spread with whipped feta. Top with sliced avocado, tomato, and your favorite burger fixings.