Preheat the oven to 400 degrees F (200 degrees C).

Heat 1 tablespoon olive oil in a pan over medium heat; add garlic and cook for 1 minute. Stir in chard and cook until soft, about 10 minutes. Add basil, salt, and pepper, and cook until basil is soft, about 2 minutes. Remove from heat and let cool completely, about 15 minutes. Place cooled chard mixture between 2 towels and press to remove excess moisture.

Stir goat cheese and chard mixture together in a bowl until thoroughly combined. Fill zucchini blossoms about 3/4 full with the goat cheese mixture, and pinch the ends closed. Place filled blossoms in a roasting pan; drizzle with 2 tablespoons olive oil, and sprinkle with salt and pepper.

Roast in the preheated oven until squash blossoms are hot, about 10 minutes.