I would doubt that the Church has any official teaching regarding the consumption of the fragment of consecrated bread placed in the chalice at the Fraction Rite.
We must remember that “the Eucharist” is Jesus Christ – His Body, Blood, Soul, Divinity and Humanity. (See the Thread entitled Only receiving the Body of Christ?) Therefore, there is no distinction between receiving the Eucharist in the form of the consecrated bread, the consecrated wine, or both at once. In fact, when the presiding minister places the fragment of consecrated bread into the chalice (often referred to as the “commingling”), he whispers: “May this mingling of the Body and Blood of our Lord Jesus Christ bring eternal life to us who receive it.”