I believe the issue is that you are not resolving the response from your request, try returning a promise and resolving only when you have your response (or error)

this.fetchResult = function(uri) {
  return new Promise((resolve, reject) => {
    fetch(uri, {
      method: 'get'
    })
    .then(response => response.ok
      ? response.json()
      : reject(new Error(response.statusText)))
    .then(result => resolve(result))
  }
}


const map = {
  'close': async () => {
     const success = await this.fetchResult(action);
   }
}

Remove the async from fetchResult there is no reason for it. You're awaiting the result of the Promise in fetchResult, you only need the async keyword if fetch result is written in async await as well:

await res = fetch(url)

Thanks @Toto Nabendo and @ggorlen!

Working solution:

export class Foo {

    constructor(el, menuItems ) {
        this.id = this.el.dataset.id;
        this.menuItems = menuItems;

        this.fetchResult = async function (uri) {

            return fetch(uri, {
                method: 'get'
            })
                .then(response => response.json())
                .then(data => {
                    return data.saved === true;
                })
                .catch(function (err) {
                    console.log('Failed ', err);
                    return false;
                });
        }

    }

    async doAction(action) {

        const uri  = 'foo/'+ action +'?id=' + this.id;

        const map = {
            'close': async () => {
                const success = await this.fetchResult(action);
                console.log(success);  
            }
            /* some other actions */
        }
        
        return map[action];
    }

    attachDialogs() {
        this.menuItems.addEventListener('click', (element) => {
            this.doAction( element.dataset.action )();
        });
    }    
}