I haven't had any severe burns in the kitchen (yet, knock on wood). but I've cut myself quite deeply slicing carrots lenghtwise, etc.

I keep a bottle of gel superglue in the kitchen, it closes the wound rapidly and it usually heals up in a week or two at worst.

I have a plethora of oven mitts, most of them are barely acceptable for blocking heat. I finally bought the silicone mitts, which you can propably use to forge steel with, but you loose all dexterity, its like unflexible mittens.
these days I use a kitchen towel and an oven mitt. but you know i wear a belt and suspenders so I like to be ready for battle at a moments notice.

also i force myself to focus on the task at hand, I have a tendency to wander into thinking about the next step in preparation or the freakin song on the radio or whatever. which makes accidents easier. I also own a collection of half a dozen thick cutting boards which double as trivets, So i can keep the hot stuff away from my prep area.

still I have a healthy respect for ovens and knifes. I started sharpening the knifes weekly, the dull knifes are the most dangerous..

glad to hear you hands are okay, and everything worked out and the family didn't complain about the smell of burnt flesh during dinner......

 

Superglue works really well. I knew a fellow who tripped heavily on mushrooms for a year and played 12 hours of classical guitar who swore by Crazy Glue to glue together where his nails meet the skin on his fretting hand. Carlos was an amazing fucking guitarist!