Make the garlic breadcrumbs: heat 1 1/2 tablespoons olive oil in a medium skillet over medium heat. Add the 2 smashed garlic cloves and cook, stirring often, until fragrant (about 2–3 minutes) and the garlic is just turning golden.
Add 1/2 cup panko to the skillet and stir until the crumbs are evenly coated in oil. Cook, stirring frequently, until the panko is golden brown. Remove from the heat, stir in 1/4 teaspoon kosher salt, transfer the breadcrumbs to a bowl, and set aside to cool.
Make the yogurt-feta spread: in a medium bowl, stir together 1 1/2 cups plain Greek yogurt, 1/3 cup feta cheese, 1 1/2 tablespoons finely chopped parsley, 1 1/2 tablespoons finely chopped dill, 1 1/2 teaspoons lemon zest, and kosher salt and black pepper to taste. Set aside.
Prepare the squash: trim the ends of 2 medium zucchinis and 2 medium yellow squash. Cut each zucchini and squash in half lengthwise, then cut each half into spears. Place the spears on a tray or in a bowl, brush all over with 2 tablespoons olive oil, and season with kosher salt and black pepper to taste.
Grill the squash: heat a grill or grill pan to medium-high. Place the squash spears cut side down on the hot grill and cook 3–4 minutes per side, until tender and well-marked. Work in batches if needed so the grill stays hot.
Assemble and serve: spread the yogurt-feta mixture on a large plate or platter. Arrange the grilled squash on top, sprinkle with the garlic breadcrumbs, and finish with crumbled feta cheese, additional lemon zest, and chopped dill as desired. Serve immediately.