Shallots: bulbs divided/multiplied

Back in late march I planted like 18 shallots I bought from the store in my raised bed.   They grew shallot greens, which I have harvested multiple times.   They became quite dense and I noticed the shallot bulb divided into like about 7 to 10 green onion shaped plants.  Photo below.

Should I beak these apart and transplant them into a new area, like 9 per square foot?   Will I get new bulbs by say end of fall?  I'd love not to have to buy shallots ever again from the grocery store but I really don't know the whole process, overwinting etc.. Seems a bit complicated.

Shallots are like garlic in that each clove can be planted separately. This page first explains growing scallions, then goes on to explain growing shallots: Shallots and Scallions

Essentially, it recommends planting 6 inches apart, and that the new bulbs are ready in 90 to 120 days.