Initially Twitter is not conceived for image shares and that is the reason at the Tweet message no parameter for image to be added by default. That makes impossible to make Tweet with image via Easy Social Share Buttons, which is really big disadvantage, because nowadays the visual part of each content is really well-thought-of.
1. Compose new Tweet at Twitter and include there the link of the post
2. Press the three dots under the tweet and choose Embed Tweet
3. Turn off the option Include media and copy the last part of the Tweet text (pic.twitter.com/Iv0N8Ra7Qk)
4. Paste it at Easy Social Share Buttons: Share Customization
Easy Social Share Buttons: Share Customization can be found at edit mode of each post.
These steps have to be followed for each new post. We know that it is a little bit boring, but we haven’t found easiest way for now. So focus over the images and enjoy the shares!