That's a slang expression for not cutting any corners. Follow EVERY single instruction.
I'm not sure where the term "to a T" came from. We(in the US) seem to say things like .... "dot the I's and cross the T's" to make sure that a printed page is proof read before anyone sees potential gramatical errors.

There's a doo-wop song by the Capri's ("Morse code of love") that uses the line ...
"...well, I dotted the I's and I crossed the T's, and now I'm beggin' pretty please ...."
Such poetry, huh?
It's a great jitterbug number that sounded like a 50's tune but was actually recorded MUCH later,(1975?) and Manhatten Transfer covered it as well.... maybe even better.

And, BTW - your English is great. I envy anyone that can converse in more than one language. I can barely get through "La Bamba" without insulting some of our Latin friends ! (NO Idea what I'm singing !)