Shimano GRX Left 2X lever with dropper post

I am assuming that you are using a 1x drivetrain and have, for some reason, a 2x left shifter (maybe after changing the chainset to 1x). If that assumption is false, you need to add a dropper lever or you won't be able to shift your gears - there are some that are designed specifically to fit near the stem on the handlebars of gravel bikes.

If you want to use your existing shifter to operate a dropper, the main problem is the indexed gear shift mechanism that keeps the cable at a specific length/tension for each gear.

A simple bodge using your current shifter would be to simply try to use the lever as-is. Then, you would have to "shift" to the position of the largest cog, operate the dropper and then shift "back down" when the dropper is in your desired position.

The next step up would be to irreversibly modify your shifter. This involves taking apart the delicate mechanisms so don't do this if value any kind of warranty or you do not feel comfortable handling small small springs, ratcheting mechanisms and whatever else you may encounter in the insides of the shifter!

You basically remove the hoods and take apart the shifter until you see the indexing mechanism, i.e. some toothed parts and pawls that 'click' into them when operating the shifter. Once you have identified the actual parts that keep the cable under tension when in the large gear, you have to grind/hacksaw away these teeth or remove the pawls. Then reassemble the shifter, taking special care of any springs inside the mechanism. As a bonus, you can remove the now-useless downshift lever if it bothers you.

The details vary with each shifter and I have not done that myself. So I sadly cannot provide any images, but searching the internet for something like "shimano left lever dropper mod" will give you an idea of what to expect.