Could extreme climate zones exist next to each other?

I would love to shape my planet in a way, that allows me to have an extremely cold, icy desert, next to an extremely hot, sanddesert.

By "next to", I don't mean a few hundred miles apart, with gradually equalising climate. I'm talking about a foot-march of, at most, 4-5 minutes between the two. Light a cigarette on the edge of the desert, be finished with it when reaching the icy desert.

Both areas should engulf roughly 140.000-180.000 square miles.

Is there a way an area like this could form naturally, without humans fiddling around with stuff and without magic?

If there is no other way around it, I'd be willing to use mountains to accomplish that, but I'd like to keep both areas as more or less flat-grounds.

You may choose the planet's positioning around a star of your choosing, its size, number of moons, all these shenanigans.

Edit: as suggested, I'll place a couple more constraints to narrow the question down a bit:

1. He who makes it plausibly possible to get rid of the height-problem (only option being vertical distance as opposed to horizontal) will be awarded 10 imaginary upvotes from me! That does not mean, answers that encourage the vertical distance are not welcome.
2. Underground-systems should, if possible, only be used, if both areas are actually underground. That makes underground-systems useless, probably, but I'll let myself be surprised.
3. Tidal-locking of a planet, if usable for this question's purpose, is obviously allowed.
4. These 2 Areas do not need to be habitable. The rest of the planet should, though. [by people that evolved there, no visitors]
5. It does not have to be water-ice. Any ice that allows for this situation to happen is welcomed. Bonus points for ice that does not kill people that want to cross it.

Edit 2: Temperatures

You may assume temperatures as follows.

Icy Desert:

• in extreme cases dropping to -90 to -120 °Celsius.

• should average somewhere between -60 to -70 °Celsius.

Border:

• anything that helps you (and therefore me) come up with a solution for the problem.

Sanddesert:

• in extreme cases rising to up to 80-100 °Celsius.
• should average somewhere between 70 to 80 °Celsius.

Should you have an idea that really could work, but needs small corrections on those numbers, add them to your answer, they'll definitely be acknowledged.