Your biggest challenge will be the interface. Different generations of laptops use different interfaces, but all are digital and basically none are compatible with DVI/HDMI (let alone DP) - or with the VESA feature connector. There exist desktop cards with interfaces for some flat-panel monitors, but they are rare - in particular cards with Chips & Technologies VGA often have them. Also quite a few industrial (mainly PICMG) boards have flat panel options.

So first question is: what kind of 'very slim desktop computer' do you have / want? If you already have it, what display hardware does it have? If you don't have any yet, what sort of period/performance are you targeting?

Note that you can get pretty cheap laptop-sized displays on AliExpress (in fact they are simply laptop or tablet screens without a SoC/CPU) that have HDMI and/or VGA input. I recently purchased a 10" full HD IPS monitor with (mini)HDMI for EUR 64. My main purpose was to have a portable display for testing purposes (at home and at the repair cafe I volunteer at), but I have considered exactly your use-case as well, perhaps connecting this to a Via Epia mITX system (in a case with VESA mount) in my case for late DOS stuff. Won't work with my specific setup as the Epia does VGA out and this monitor only does HDMI in, but there are others that do have VGA.

No, I mean a vintage laptop, like a Compaq Armada or Toshiba Satellite with the screen removed.

Edit: something like this.

So take an old laptop, remove the screen and then use the base that is left as a mini desktop, connected to a monitor?

What do you want to be able to run on the system? Will it be plain DOS? Windows? (and what version?) Do you want to play games on it?

A lot of older machines are not sensitive to the screen being broken/removed, and will just work with video on the VGA connector (if it has one!); it depends on what you want to do with it.

I was thinking of using it as a machine for writing. It's not particularly suited for gaming, even though they do run.

The laptop in question is a Toshiba Satellite 1805-S273 with Windows 98SE

Are you using an external keyboard or the laptops keyboard?

If using internal, you may care about the battery, some are basically a whole corner or back of a laptop and leave it unstable when missing, some may require that the laptop is electrically (jumper wire) or electronically modified to boot with no battery at all.

If you are using external keyboard, it may be of no benefit to remove the screen if the laptop can be set to run with screen closed as many can. It may not be thinner to slide under a short shelf if the hinge mounts stick up the depth of the screen. Often indicator lights are built into the base of these and it can be integral to case top. Anyway, lumpy bits or fugly holes might be hard to avoid. In some instances you may lose speakers, if post 2000, you might lose wifi antenna.

Relative easiness might be deceptive as some may disconnect mechanically without opening main case, just popping off covers and unbolting, but to disconnect cable might need complete disassembly.

Good advice to consider. I've found late 90s Toshiba laptops to work fine without the battery. So far.