How to compose two async function in Rust

I'm not sure it is possible to do it that simple with impl Trait-s. The one solution I can come up with is old-fashioned future types usage without async-await feature. TLDR: full playground. Async-await uses a generators which internally holds a state machine, so we need to define it manually:

enum State<In, F, FutOutF, G, FutOutG> {
    Initial(In, F, G), // Out composed type created
    FirstAwait(FutOutF, G), // Composed type waits for the first future
    SecondAwait(FutOutG), // and for the second
    // here can be a `Completed` state, but it simpler
    // to handle it with `Option<..>` in our future itself
}

Then define a composed type itself:

struct Compose<In, Out, F, FutOutF, G, FutOutG> {
    state: Option<State<In, F, FutOutF, G, FutOutG>>,
    _t: PhantomData<Out>,
}

// And "entry-point" would be something like that:
fn compose_fut<In, Out, F, FutOutF, G, FutOutG>(
    i: In,
    f: F,
    g: G,
) -> Compose<In, Out, F, FutOutF, G, FutOutG> {
    Compose {
        state: Some(State::Initial(i, f, g)),
        _t: PhantomData,
    }
}

Then comes the most complex part - impl Future itself, here a base impl declaration without implementation:

impl<In, Mid, Out, F, FutOutF, G, FutOutG> Future for Compose<In, Out, F, FutOutF, G, FutOutG>
where
    FutOutF: Future<Output = Mid>,
    F: FnOnce(In) -> FutOutF,
    FutOutG: Future<Output = Out>,
    G: FnOnce(Mid) -> FutOutG,
{
    type Output = Out;

    fn poll(self: Pin<&mut Self>, cx: &mut Context) -> Poll<Self::Output> {
        // here comes the magic
    }
}

Values transformed as following: In -> Mid -> Out, where F and G are our composed functions and their output are FutOutF and FutOutG accordingly. And finally Future::poll implementation:

let this = unsafe { self.get_unchecked_mut() };

let state = this.state.take();

match state {
    None => Poll::Pending, // invalid state
    Some(State::Initial(i, f, g)) => {
        let fut = f(i);
        this.state = Some(State::FirstAwait(fut, g));
        cx.waker().wake_by_ref();
        Poll::Pending
    }
    Some(State::FirstAwait(mut fut, g)) => {
        let val = match unsafe { Pin::new_unchecked(&mut fut) }.poll(cx) {
            Poll::Ready(v) => v,
            Poll::Pending => {
                this.state = Some(State::FirstAwait(fut, g));
                return Poll::Pending;
            }
        };
        let fut = g(val);
        this.state = Some(State::SecondAwait(fut));
        cx.waker().wake_by_ref();
        Poll::Pending
    }
    Some(State::SecondAwait(mut fut)) => {
        match unsafe { Pin::new_unchecked(&mut fut) }.poll(cx) {
            Poll::Ready(v) => Poll::Ready(v),
            Poll::Pending => {
                this.state = Some(State::SecondAwait(fut));
                Poll::Pending
            }
        }
    }
}

I avoid any library to make it "plain", usually unsafe parts are handled with pin-project or futures::pin_mut. The state management is fairly complex, so I suggest to re-check the implementation, there might be mistakes.