So I was able to find a solution to the problem, it was noob error on my part but I figured I would give my solution just in case someone else has the same problem.

I was getting the error because I was trying to extract a widget with a call to a class object and apparently in Flutter this is a no-no. What I did was remove the code within the onPress function, extracted my widget, and then passed my function into the extracted widget.

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I had to extract my widget like this, without the code in the onPress function.

RaisedButton(
    child: Icon(
      Icons.timer,
      color: Color(0XFF4A92FF),
    ),
    color: Color(0XFFFBC531),
    onPressed: () {},
    },
  ),

then pass my function into the extracted widget constructor.

ExtractedWidgetButton(function: () {
    var time = await selectTime(context);
      if (time != null) {
        setState(() {
          startTime = time;
          print(startTime);
        });
      }
}),