Step 1Cut the onion in quarters. Cut one quarter into a quarter-inch dice and reserve for serving. Grate the remaining onion on the large holes of a box grater.
Step 2Heat the oil in a large skillet over medium-high heat. Add the grated onion and half a teaspoon salt. Cook, stirring occasionally, until the onion is translucent, 5 to 7 minutes. Add 8 ounces of the “beef,” sprinkle with salt, and cook, smashing it into tiny bits in the pan, until browned, about 5 minutes. Add the chili powder and mushroom powder, if using, and cook, stirring, for 30 seconds. Add the tomato paste and cook, stirring, for 1 minute. Add one-fourth cup water and cook, stirring, until saucy. If the mixture is still thick, add more water, 1 tablespoon at a time, until the consistency of loose chili sauce. Season to taste with salt and keep warm over low heat.
Step 3Form the remaining pound of “beef” into eight patties (2 ounces each) the same diameter as the hamburger buns; the patties should be thin. Heat a griddle or two cast iron skillets over medium-high heat. Coat with a thin layer of oil, then add the patties. Cook until the bottoms are browned, 3 to 4 minutes, then flip the patties. Sprinkle the tops generously with salt and continue cooking until the other sides are deeply browned, 3 to 4 minutes.
Step 4While the patties cook, smear mustard on all the toasted sides of the hamburger buns. Divide the chili sauce among all the bottom buns, then top with the patties, pickles, diced white onion and tomato slices. Sandwich with the top buns and serve immediately.