First, preheat the oven to 190 C / 170 C fan oven / 375 F.
Prepare the shortcrust quiche crust. You can make it yourself, or you can buy ready rolled shortcrust pastry. Roll the pastry out with a rolling pin, then place it in a quiche tin. Make sure to press it gently into all corners and up along the edges of the tin. Prick the bottom of the quiche with a fork a few times, then set it aside.
Then, prepare the vegetables. Dice peppers, red onions and courgette into bite sized pieces, and cut the cherry tomatoes in half. Add these to a baking tray, and drizzle olive oil on top. Rub the olive oil over the vegetables.
Place the vegetables and the pastry in the oven, and bake them until the vegetables are soft and roasted, and the pastry is baked through, about 20 minutes. Remove them both from the oven, and leave them to cool down slightly.
In a bowl, whisk together the eggs, cream, salt, pepper and basil. Add the roasted vegetables, and pour the mixture into the pastry case. Add diced feta cheese on top, then place the quiche in the oven.
Bake the roasted vegetable and feta quiche until the egg mixture has solidified, about 30 minutes. Ensure that the quiche is not wobbly at all before removing it from the oven.