Roll the puff pastry out so that it's large enough to fit in your removable bottom tart pan. Then press it into the tart pan ensuring that the edges are covered and wrapped slightly around the edge to ensure it doesn't slip. Then using the tines of a fork, press holes into the puff pastry so that air can escape while it bakes. Pre-bake the puff in the oven for 5-10 minutes. When you remove from the oven, stamp out any air bubbles that may have arisen during cooking.
Bring a non stick pan up to medium heat and melt the butter. Add your onions, leeks, green onions, shallots, and garlic. Cook over medium-low heat stirring occasionally, ensuring that the onions aren't burning but slowly browning. This process should take about 10 minutes.
While the onions are cooking, whisk together your creme fraiche, eggs and spices.
When the vegetables are close to being done, throw in your broccolini and allow to cook for 1-2 minutes longer so that the broccolini stems soften. Remove the onions from heat.
Layer the browned onions on the bottom of the tart first, ensuring that they are spread evenly. Gently pour over egg mixture on top and jiggle with your spoon to get it spread around the veggies.
Place on a baking sheet and in the oven to cook for 20 minutes.
Remove from heat, let cool for 5 minutes in mold, then pop out when you're ready to serve.