I give her stuff in big enough pieces that she can hold it in her fist and suck on the end of it. From what I've read, a baby's gag reflex is much further forward than ours so they will spit something out before they can choke on it. You might get a bit of gagging, but that's perfectly normal and not dangerous. It's only if they go very silent that you need to freak out.
Iz has hardly gagged at all, bit of coughing occasionally but I've honestly not had a single scary moment and she just spits out anything she can't handle.
I think even if you're doing purees you're meant to start introducing finger foods quite soon after 6 months for developmental reasons.
She has normal porridge in the mornings, not baby porridge. I've given her toast, melon sticks, green beans, cucumber, cooked carrot sticks, broccoli florets, pancake, chicken, ricecakes topped with all sorts of stuff, mango, plums (skinned), pasta (twirls are easy to pick up, spaghetti is hilarious) all sorts of stuff. She can't always eat much of it, but she really enjoys trying to pick it up and giving it a good suck. And she's really good at feeding herself with the spoon now.
I wouldn't give raw carrot as if she did manage to break a bit off she wouldn't be able to get it soft to swallow. So I'd worry about that being a choking hazard. But anything she can suck soft or mush with her gums is fair game.