Q:

Dojo: TabContainer - how to place menu button in the header?

is it possible to place a button in TabContainer header on the left side ?
I want to place it next to First Tab.
Thanks for help :)

A:

You are going to have to create your own tab controller widget.  The steps would be as follows:

Create MyTabController that extends dijit.layout.TabController
Create a template for MyTabController that has a place for the button
Update MyTabController javascript to create the button

You can use your new controller widget in one of two ways.  If it were me, I'd also create my own tab container widget that extends dijit.layout.TabContainer and overrides the _makeController function to instantiate the new controller.
Alternatively, you could pass in the _makeController function when instantiating the TabContianer widget
var tc = new dijit.layout.TabContainer({
    _makeController: function(srcNode) {
      ...
    }
}, node);

You can look at the dijit.layout.TabContainer source to see what needs to be done in the _makeController function.

