Brushes (which can also be applied to pen strokes, BTW) have colour limitations baked in to how they work: you can over-ride their pre-stored colour to allow tint shift or hue shifts or both; they will not accept gradients. To get into the settings of a brush you've applied, open your applied brushes pane and double-click the brush in question - go to the colourization area, and you can change there to accept tints and shade, or hue shifts. This, as already stated, still won't allow you gradients with applied brushes.
However, you are not actually stuck: Illustrator allows you to draw simple strokes with the pen tool, and then afterwards using the Width Tool you can edit the thickness of a pen stroke to get an imitation of carefully-controlled pressure-based width variations - and as there isn't a Brush per se applied to the stroke, this stroke will accept gradients!
So it's not exactly what you asked for, but it can get you a close visual analogue to the type of brush look in your example image, and with a gradient, and still an editable path - like so.
Actually I was in error here - the basic round brush will accept gradients too, and you can set the size attribute to accept pressure as an input - see below.
So, to recap what we've learned - you've two viable options, both of which are quite effective:
1. the basic brushes, including calligraphic brushes, with pressure driving size
2. pen stroke with the Width Tool after the fact, and gradients will work fine for either
You cannot use art brushes or pattern brushes and gradients on strokes.
Hope this helps.