Custom component similar to DatePicker

I need to create a component that would consist of two controls:
TextInput and an icon. Something looking and working similar to a
DateField. But when I click on an icon I will show a list to choose some
values from. I want that list being shown below the text field - just like
a calendar is shown. Where would I start developing something like this? I
want my component working in the same way DatePIcker works.

Thanks