The Tillamook cheddar burger is a regional burger whose defining choice is a cheese that refuses to behave like the slice most burgers use. American cheese is engineered to flow, slumping into the seared crust and sealing to the patty as one molten layer. An aged Oregon cheddar does the opposite: it is a real cheese with structure, so it softens and slicks at the edges but holds its body rather than dissolving, and it brings a sharp, slightly granular tang the processed slice was formulated to remove. That trade, melt behavior given up for flavor and bite, is the entire argument of the sandwich. The rest of the build is the plain frame around the one cheese decision.
The craft is in handling a cheese that does not self-seal. Because aged cheddar will not fuse the patty the way an American slice does, it goes on while the beef is still on the heat and is often given a brief cover so trapped steam softens it down over the meat before it sets. The patty is kept thick enough to carry a cheese with this much presence without being overwhelmed, and the bun is soft and faintly sweet so it compresses to the stack rather than fighting the firmer cheese on top. The cheddar's sharpness changes what the rest of the build has to do: where a milder slice needs pickle and raw onion as the acidic counter, the cheese is already supplying part of that cut itself, so the cool, crunchy accents read as support rather than rescue. The dairy-pride framing on the bun is doing real work rather than decoration: a cheese chosen for character instead of melt is the whole reason this is a named build rather than a generic cheeseburger.
There is not a wide spread of codified mutations here, because the variation is mostly the cheese itself. A sharper or more aged block pushes the tang further; a thin layer of a melting cheese under the cheddar is sometimes used to recover the seal the cheddar gives up. It sits inside the broad American burger family, where the patty is the fixed point and the choice on top is the argument, alongside the green chile, butter, and regional cheddar and bison builds. Those each deserve their own article rather than being crowded in here.