Wash and drain broccolini and herbs. Preheat your grill to medium-low heat.
Remove tough stems from herbs and add to the bowl of a food processor or blender with the sundried tomatoes, a pinch of salt and freshly ground black pepper. Process until ingredients are chopped fine. Taste and adjust seasoning - it should be very herby and noticeably salty.
With the motor running, slowly add the oil to the herb mixture, processing until smooth. It may take a minute or so to finely process the sundried tomatoes.
Brush chicken on both sides with half of the herb oil. Toss the broccolini with the other half of the oil, and arrange broccolini on a grill pan.
Place chicken directly on the grill grates, alongside your grill pan. Grill until broccolini is tender and crispy in places, stirring occasionally, about 20 minutes. Grill chicken breast until the internal temperature reaches 160º, about 10-15 minutes, flipping only once.