There are approximately 250 quasars discovered at redshift $z\geq6$, of which only a handful were detected in radio bands, and even fewer were imaged with the highest resolution very long baseline interferometry (VLBI) technique. Here we report the results of our dual-frequency observations with the Very Long Baseline Array (VLBA) of two such recently discovered quasars, VIKING J231818.35$-$311346.3 at $z=6.44$ and FIRST J233153.20$+$112952.11 at $z=6.57$. Both extremely distant sources were imaged with VLBI for the first time. The radio properties of the former are consistent with those of quasars with young radio jets. The latter has an UV/optical spectrum characteristic of BL Lac objects, of which no others have been found beyond redshift 4 so far. Our VLBA observations revealed a flat-spectrum compact radio source.