Preheat the oven to 180 °C (356 °F).
Prepare the broccoli: if fresh, wash and cut into bite-sized florets; if frozen, keep as is. Bring a large pot of water to a boil and cook the broccoli for 5–7 minutes, until slightly tender. Drain thoroughly and shake off excess water; pat dry with kitchen paper if needed.
Grease an ovenproof dish with the 1 tbsp olive oil.
Spread the drained broccoli evenly in the greased dish.
Sprinkle all 300 g of the cheese evenly over the broccoli.
In a bowl, whisk together the 3 eggs, 200 ml yogurt or sour cream, and 1 tsp salt until smooth.
Pour the egg–yogurt mixture evenly over the broccoli and cheese.
Add the thyme: if using fresh, strip the leaves from the stems and scatter the handful of leaves over the top; if using dried, sprinkle 1 tsp dried thyme evenly.
Bake in the preheated oven for 20–25 minutes, until the cheese has melted, the mixture is set, and the top is golden and lightly crisp.
Remove from the oven and let rest for a few minutes before serving.