I read the article about how impossible it is to make a real burger. And about all I can say about THAT is that if you are going to use a cheeseburger as the gold standard recipe, and all the stuff you put on it identifies it as a California burger, it might be a good idea to try it in California first before declaring that it is impossible.
I am sure the typical hamburger can easily be made in the fall.
Although I don't have a steer to slaughter this fall, I am sure people could butcher a goat at any time. Of course, with goats being about the only local livestock, one would expect an abundance of goat milk in the spring and summer. No need to wait a year to harvest some rennet. Goat cheese can be made in a couple of hours with plain ole goats milk and some lemons. Now if you insist on that square orange stuff, there is no hope for you anyway, and it is probably best that it is impossible for an average person to make just so you don't hurt yourself.
Cucumbers come into harvesting from June to September, giving any cook ample time to make pickles and store them hassle-free until the rest of the ingredients are ready.
Lettuce grows extremely well in the fall, and people here harvest lots of lettuce right about the time when the last of the summer tomatoes are harvested. Of course, the tomatoes can be picked and the vine hung up right out on the patio, and you can have a fresh sliced tomato up to and including New Years day. (Ask me how I know this....) If that isn't good enough, you can grow tasteless winter varieties that will mimic the one they put on your fast-food meal perfectly.
Mustard is just about the easiest condiment to make. My mustard seeds would be ripening by now, had the goldfinches not stripped the plants clean of all seed pods and flowers. If I really had to have mustard, I would have planted more. If I hadn't gotten around to fermenting last fall's apples into vinegar, I'd just make mustard with sauerkraut or pickle juice. It works really well. The reason I didn't get around to planting more mustard for seed is that I grow plenty of hot mustard plants in the fall, and a nice hot mustard leaf can do double duty on the hamburger instead of the mustard and the lettuce. And it is much easier since after the goldfinches get into all the seed, they toss it all around and it reseeds itself everywhere and all I have to do is sit around and marvel at the efficiency of laziness.
The hardest thing to duplicate is the bun, but that is easy for me since I don't eat that stuff much anymore and would just leave it out entirely. But in earlier times I have actually planted, harvested, winnowed, ground and baked wheat. Believe me, the people who say our diet should be based on healthywholegrains and other seed foods just don't know diddly-squat about growing and securing their own food. It is really hard to grow seeds. You might plant a whole row of something, and then have mice come by and take much of it at the last minute, or have it shatter all over the place. Even for something super easy like amaranth, gathering even a cup of the seeds takes many hours and it tastes like red poi. (Ask me how I know this...) There is good reason why the Chumash put a much higher price on chia seeds than on acorns. If you have actually tried to harvest chia or basil seed or acorns, you will immediately notice the difference in time needed to get a cup of seeds vs. nuts.
Around here, I might be able to make a steamed pile of acorn flour, and this would take quite a bit of time. I don't know how it would taste because every time I attempted to make food from acorns, it was a failure. There is a good reason why the Chumash acorn-grinding stones were near the ocean. They could have just grabbed more seawater for soaking, bypassing the moldy stage that my acorn mess got into. And they did. Acorns made up around forty percent of their diet.
Actually, the hardest food around here to replicate in a fast food meal is the potatoes. They only grow in the spring, and are usually rotten by summer. I grow a few of them, but mainly it is "you want sunchokes with that?" And if you want to supersize THAT, well there is still no hope for you.