Short answer, yes. The side shoots won’t be as big, but they will come. You might also try starting some broccoli now for fall harvest. Don’t know your zone but here I’m about 7-8 and I can have broccoli into November.
Most varieties of broccoli will send out side shoots of new florets after the main floret is cut. They will be smaller in size however. But, they will taste just as good as the large center floret. Some varieties are more prolific in side shoot production than others. I would save the plants, as they make a great fall crop. They may also overwinter to give you some excellent eating as soon as the snow melts next spring! Next spring you can let them go to seed and save the seed for the next crop.
Your broccoli should continue producing side shoots for weeks. Some varieties produce more side shoots than others. The floret blossoms grow to the same size as those in the original heads, but it takes a number of shoots to equal the size of the first heads. In FL I can harvest side shoots for a couple of months, but they diminish in size with each cutting.