Begin by washing and trimming the leeks, focusing on the white and light green parts. Quarter them lengthwise, rinse thoroughly to remove any dirt, then thinly slice. Peel and finely dice the carrots and red bell pepper. Dice the celery into small pieces to match the size of the other vegetables.
Heat 1 tablespoon of butter in a skillet over medium heat. Add the diced celery, sliced leeks, carrots, and red bell pepper. Season with a pinch of salt and freshly ground pepper. Sauté gently for 6-8 minutes until the vegetables are tender but still retain a bit of bite. Remove from heat and let cool slightly.
In a medium bowl, combine ½ cup heavy cream, 1 whole egg, and 1 egg yolk. Whisk together until smooth and fully blended. Add a pinch of salt and pepper to season.
Preheat your oven to 350°F (175°C). Spread the sautéed vegetables evenly over the partially baked and cooled 9-inch tart shell. Sprinkle ½ cup grated Gruyère cheese over the vegetables. Carefully pour the egg and cream custard over the filling, allowing it to fill the gaps.
Place the tart pan on a baking sheet and bake in the preheated oven for 30-35 minutes. The quiche is done when the custard is set around the edges but still slightly jiggly in the center. The cheese should be melted and golden.
Allow the quiche to cool for 10-15 minutes before slicing. This resting time helps the custard finish setting and makes slicing easier. Serve warm or at room temperature.