Beginner broccoli

I'm my zone 9b broccoli is a winter crop. I don't know why I have never grown it before, we love it. I thought it was like head lettuce, one and done. Last fall I planted a bunch of broccoli. WOW I was totally wrong. It was super easy to grow. Once I cut the beautiful large head of broccoli it kept growing a lot of little side shoots.  I over planted an didn't pick fast enough, and it all started to flower.  Still we got lots of broccoli, and it was getting to warm for broccoli anyway.  
I cut the stalks off about an inch from the soil.  I didn't want to disturb the soil by pulling it out.  We have been having a very mild spring. It's only now starting to get hot.  Instead of dieing back the broccoli is regrowing.  My first thought was cool we will see what happens.  But then I was reading something, and it was saying broccoli is a very heavy feeder, making it a crummy companion plant.  Now I don't know if I should cut it back to soil level, leave it be, what???

Maybe let it flower and save more seeds? Will it be a perennial in your area?

Your lettuces grow so well by the way.

Thank you. I have no idea if it's a perennial in my area???. The lettuce looks good, but it's so bitter no one but the chickens eat it. I have tried different varieties, made sure to plant when the weather won't get above 80, water lots, and still very bitter. Lettuce is one of the simple veggies I just can't grow successfully.

I'm in zone 8 and I had a broccoli live and produce florets for more than five years. It just died this spring. 😭 RIP My dear broccoli.

I say just leave it if you want to try out. I cut back the stalks everytime they flowered and went to seed. If I didn't, they just died back a little bit kept producing new shoots from the bottom six inches. It never produced viable seed but I think that's because it didn't have any other broccoli around. My kids would have sword fights with the old wooden stalks when I'd cut them off but the little florets were tender and yummy.