JS is a single threaded language, so there's just one long line of instructions, there's no actual background tasks.
A promise just runs at the next available slot the event loop provides so it's just as soon as it can run.

In this case I'm guessing other part of the code running in your JS engine is blocking the event loop preventing your code to run just after one second.

Where are you running it? Browser or server?
Are you using a framework? If so, have you tried just this piece of code without any other imported code?

Here is the code for anyone else who want to try

async function slowLoop() {
    while (true) {
        console.time('sleepDuration')
        await new Promise(resolve => setTimeout(resolve, 1000))
        console.timeEnd('sleepDuration')
    }
}

It is not suppose to deviate too much. setTimeout will execute whenever possible at / after the given timeout duration. I am getting between 1006 to 1023 on my sysyem

Is your device performance starved during this test?

I don't know anything about JS, but those timings are suspiciously close to integer multiples of your sleep duration, after the first.

Based on what you've shown, "All over place" is not what I'd call it. It either fires "on time" or takes "exactly-ish" twice as long.

Pretty consistent defining/running directly in the console on a blank tab in Firefox

It’s the while loop that may be blocking the event loop and slowing the execution.