Absolutely not. What this EMHC is doing essentially is giving benediction with the Eucharist, a liturgical action that is reserved to a priest or deacon. The document Holy Communion and Worship of the Eucharist Outside of Mass states:
[Special] ministers [of Communion] may open the tabernacle and also, as required, place the ciborium on the altar or place the host in the monstrance. At the end of the period of adoration, they replace the Blessed Sacrament in the tabernacle. It is not lawful, however, for them to give the blessing with the Sacrament. (HCWE 91)
I recommend taking to your pastor a copy of this document, which can be found online at the Web site for the Diocese of Fargo (www.fargodiocese.org), and explaining to him your observation that one of the EMHCs is giving Eucharistic benediction during the distribution of Communion.