Yes, of course, you can do some basic mountaineering while on Everest Base Camp Trek. There are several famous trekking peaks in Nepal that you can experience climbing them as per your comfort. Island Peak climbing, Lobuje East Peak climbing, Mera Peak climbing, etc. are some mountaineerig packages you can book with us.
On the other hand, doing mountaineering requires some skills to be learned, so it is time-consuming as well. Moreover, without the previous experience of trekking in the Himalayas and peak climbing, mountaineering may turn to be dangerous, too. Therefore, you must consult with the experts sand trainers before trying out for mountaineering.