Preheat oven to 375°F.
Remove the frozen tart shell from the freezer. Bake the tart crust for 15 minutes so that it doesn't get soggy when you add the filling.
While the crust bakes, make the filling: Lightly beat the eggs, then add the mozzarella to the egg mixture and stir to combine. Set aside.
Melt 1 TBSP butter over medium heat. Sauté onion and broccoli until onion is soft and broccoli is bright green. Remove from skillet.
In the same skillet, melt an additional 2 TBSP butter over low medium heat. As soon as the butter melts, whisk in the flour and stir until a thick paste forms.
Gradually whisk in the milk and continue whisking until the paste becomes thick and smooth. This is your roux. Season with salt and pepper, then gradually whisk the egg and cheese mixture into the roux. Adding the eggs gradually prevents them from scrambling. Keep stirring until quite thick and creamy, a few minutes more.
Finally, add the broccoli and onion back to the skillet and stir to combine. Remove the skillet from the stove.
When the crust is done, remove it from the oven. If it has bubbled up at all, poke it gently with a fork so that it settles down.
Pour the broccoli filling into the crust, making sure not to go over the top edge. Top it with a generous blanket of shredded cheddar cheese.
Place the filled tart shell on a sheet pan (in case of spillage) and bake the quiche for 30 minutes until cheese has browned and the filling has domed slightly on top.
Remove from the oven and let cool on a wire rack. The filling will settle down as it cools Place the tart pan on a small overturned bowl or cup and let the outer ring fall off.
Let cool 10-15 minutes before slicing. Serve warm or at room temperature.