Wire nut tie together for a #4 braided aluminum wire?
He'll need to use a pull box to house this

Splicing of wire that fat can't be done in a regular junction box due to the fact regular junction boxes are too small to let the wire be bent in conformance with bending radius specifications.  Instead, what's needed is a NEMA rated (NEMA 1 if it's inside, NEMA 3R if it's outside) pull box, as these are available in dimensions that conform with the specifications in NEC 314.28(A)(2):


  (2) Angle or U Pulls, or Splices. Where splices or where angle
  or U pulls are made, the distance between each raceway entry
  inside the box or conduit body and the opposite wall of the
  box or conduit body shall not be less than six times the metric
  designator (trade size) of the largest raceway in a row. This
  distance shall be increased for additional entries by the amount
  of the sum of the diameters of all other raceway entries in the
  same row on the same wall of the box. Each row shall be calculated individually, and the single row that provides the maximum distance shall be used.
  
  Exception: Where a raceway or cable entry is in the wall of a box or
  conduit body opposite a removable cover, the distance from that wall to
  the cover shall be permitted to comply with the distance required for one
  wire per terminal in Table 312.6(A).
  
  The distance between raceway entries enclosing the same
  conductor shall not be less than six times the metric designator
  (trade size) of the larger raceway.
  
  When transposing cable size into raceway size in
  314.28(A)(1) and (A)(2), the minimum metric designator
  (trade size) raceway required for the number and size of
  conductors in the cable shall be used.


Note that for large raceways, this can lead to rather large boxes, so it may be best to mount the box to one side of the panel and leave a bit of slack length in the too-short run instead of mounting it above the panel, although mounting it below a bottom-fed panel may not be an issue, depending on the depth of the box.

4AWG Wirenut Not Found

Wirenuts are generally not made in large enough sizes to splice 4AWG wires, so your electrician will be using a different splicing device -- a mechanical setscrew connector, or Polaris&trade; connector as it is often called.  These, like wirenuts, are one-piece splicing units, but instead of twisting the wires together, each wire is individually attached using screws that clamp the wires down into a block of plated aluminum, much like a breaker box neutral or ground busbar.  While they are available in different configurations (one-sided, two-sided, in-line), most are like the picture below in that the screws sit under the caps on top while the wires go into the ports on the front and/or back (or each end) of the connector.  Furthermore, as a result of their construction, they can handle a variety of wire sizes (even in a single connector -- the smallest ones go from 14AWG to 4AWG, and the next smallest size starts at 6AWG and goes to 3/0 or more), and can handle any mix of copper and aluminum wire.