Standard and custom strings can't be combined into a single string format. This would need to be done in two steps:
var weekday = my_date.ToString("ddd");
var day = my_date.ToString("d");
var both = string.Format("{0} {1}", weekday, day);
In XAML, this would be more verbose, but the following should work (this is untested):
<TextBlock.Text>
<MultiBinding StringFormat="\{0\} \{1\}">
<Binding Path="my_date" StringFormat="ddd" />
<Binding Path="my_date" StringFormat="d" />
</MultiBinding>
</TextBlock.Text>