Lab procedures exist to determine the DNA or RNA sequences of a virus. They aren't used for ordinary screening, but they are used in research situations, or investigations like this one. The CDC report I linked to in post #3 describes what they did:
Specimens for WGS were collected during May 26–June 12; all 18 positive specimens with detectable virus (cycle threshold value <32) were sequenced using ClearDx instruments (Clear Laboratories), Oxford Nanopore MinION sequencing technology, and SARS-CoV-2 ARTIC V3 protocol for amplicon sequencing.† Consensus genome assembly was performed in Terra using Titan Clear Laboratories workflow.§ All sequences generated were classified as the Delta variant. A phylogenetic tree was constructed using the UShER pipeline and visualized using Auspice.us¶ (4) (Figure 3). Eleven sequences were genetically indistinguishable from one another; seven sequences contained additional single nucleotide variations. Among the indistinguishable specimens, six were from students of the index patient, four were from students in the separate grade, and one was from a sibling of a student in the index patient’s class, suggesting that infections occurring in the two grades likely were part of the same outbreak.
So they detected the virus using the normal PCR test, notably with less than 32 PCR cycles, and then they sequenced the viruses. That lets them determine ancestor descendant relationships in the same way DNA sequencing is used to identify human offspring or relatives.
That is how the CDC determined that the teacher had the Delta variant.
But the more important point of this article is unchanged regardless of what COVID variant the teacher had, or really if they even had COVID at all. The important point is that masks on kids in elementary schools don't do anything to stop airborne viruses from spreading. That experimentally determined result from this classroom should by itself end all mandates for kids to wear masks in classrooms.