Steven Zahm@ Casey
The connectionsAlphaEntryList() function never existed in Connections, but judging from the array arguments that is was just a wrapper for connectionsList() function. Right now, the equivalent is cnShortcode_Connections::shortcode(). This is just transitional thought and will likely break/not exist in a future update.
You’re probably better off using the core WP function do_shortcode() for now. Using do_shortcode() you would have a call that looks something like (untested):
echo do_shortcode( [connections first_name=' . $letter . ' order_by='first_name|SORT_ACS' 'template'='profile'] );
The display_profile was also never a template that existed for Connections. So maybe it was custom created or you meant profile, which does come with Connections. In the example above I used the profile template.
Hope that helps!