Look into twitter cards.
The trick is not in the button but rather the page you are sharing. Twitter Cards pull the image from the meta tags similar to facebook
sharing.
Example:
<meta name="twitter:card" content="summary_large_image">
<meta name="twitter:site" content="@site_username">
<meta name="twitter:title" content="Top 10 Things Ever">
<meta name="twitter:description" content="Up than 200 characters.">
<meta name="twitter:creator" content="@creator_username">
<meta name="twitter:image" content="http://placekitten.com/250/250">
<meta name="twitter:domain" content="YourDomain.com">