Lol. Yep.
Roux or Béchamel Sauce
Which is what you described above.
When it calls for mushroom, just add mushrooms or use mushroom stock. Chicken, ditto. Celery, celery or veggie stock.
SERIOUS TRICK:
When making a roux... Don't add the liquid right away. Cook the flour in the oil for a few minutes (until it turns cream, beige, tan, caramel, peanutbutter, mahogany, chocolate color). The lightest colors don't add flavor, but the cooked flour no longer TASTES like flour, and the taste 'cooks out' in 2 mminutes of stirring, instead of 25 minutes of cooking. The lighter colors also thicken A LOT (dark colors not as much). When I make jambalaya, I use a peanutbutter roux. When I make gumbo, a chocolate roux. Those both add incredible depth of flavor in their own right... Although the more you cook them, the less they thicken.
You can use ANY amount of oil/flour... Just keep the ratio the same. 1tbs:1tbs or 1gallon to 1 gallon... It doesn't matter. As you're adding liquid... Add slowly (it thickens almost immediately, but not instantly. Keep adding liquid until its exactly as thick as you want it).
That takes the guessing out (you know how thick its going to get, no waiting 25 minutes and then adding more flour because its thin, and waiting another 25 minutes), and shortens the cooking time a ton.
The types of oil/flour alter the recipe flavor significantly.
Butter
Lard
Bacon grease
Olive oil
Corn oil
Veggie oil
Turkey drippings
Beef drippings
Chicken drippings
Etc.
All change the character. When cooking Italian, Spanish, or Mediterranean... Olive oil. When cooking Mexican or Southern, lard (or corn oil). When cooking meat based (like beef pot pie, or chicken casserole, or biscuits n gravy)... Using meat drippings creates a WOW dining experience.
You can also start getting jiggy with flour choices. I usually just use unbleached wheat. But try using rye flour, pumpernickel, oat, rice, etc... For totally different flavor profiles.
These all add 10 minutes to cooking time (why our grandmothers fell in love with condensed soups)... But they're SUPER easy.