Quiche filling sets like an Aero bar?

Are you just using eggs for your filling? Most quiche fillings are a custard, with milk, cream, or other dairy as well as eggs, and often cheese as well.

It sounds like you are over beating your egg or custard mixture, and inadvertently incorporating air.

In general you want minimum air in a quiche filling, so whisk or stir it briefly until it comes together, but not extra.

Its harder to tell from the description, but you may also be over baking the quiche. You want to cook it until the center just barely jiggles, about 180 F / 82 C.

In my eyes, this is a very clear case of overbaking.

As SAJ14SAJ says, a quiche is basically a custard. The eggs in a custard work the same way no matter if you are making a quiche, a creme caramel, or a creme patisserie. They have lots of different proteins, which float free in the liquid part of the egg while the eggs are raw.

When you start heating the egg mixture, the proteins start to bind together, forming a three dimensional mesh. The more you heat it, the more dense the mesh becomes, because some of the proteins which did not form cross-connections at, say, 60 degrees Celsius, start forming them at 70. In the beginning, this means that your egg mixture goes from liquid to a dense liquid to a soft amorphous solid. Then, with more connections formed, the solid mesh grows stronger. The new cross-links also reduce the volume of the holes of your mesh.

At some point, you get the result you describe. The mesh is very strong, so that you perceive it as rubbery when you bite on it. It has also contracted considerably, expelling the more liquid/watery parts of the egg, while the proteins stay in the mesh and the fats cling to it. This expelled liquid also started cooking, creating bubbles large enough to see. This is where your "aero" description comes from. Depending on how you treat your quiche before cutting, and also whether there are ingredients present which can soak up this watery liquid, you can see the water flowing out of the bubbles after cutting it, forming a puddle on the bottom of the plate, or it may have disappeared (soaked up, flowed out, evaporated) before cutting.

The solution is to treat the quiche the same way you treat any other custard. Bake it on a low temperature, and take it out as early as possible. The best way to know for sure is to stick a thermometer into the middle and to remove it at around 85 degrees Celsius, but as quiches tend to give you some more leeway than other custards, you can also try learning to rely on a combination of cooking time and intuition, which works as long as you use the same oven and the same quiche pan. (The correct time will vary with differences in oven calibration, pan material and pan geometry). Covering the top with foil might be a good idea if you are afraid that the top layer gets heated too quickly, just remember to uncover it at the end for a golden crust. As a quiche has a nice isolating crust, the extreme measures needed for other quiches like a water bath shouldn't be necessary.