Moving up too fast???

Hello all! I'm new to the site and just want to thank everyone in advance for reading this!
Here's my situation/question:

I've been in the restaurant industry for 6 months now, and a line cook for 4.
I am very enthusiastic about working in the kitchen and am always doing everything I can to learn the trade while making a solid effort to go above and beyond anything that is asked or required of me.
I will be straight with this. I want to run a kitchen.
I've been at this particular restaurant for 1 month now. All new hires/cooks have years of experience, but not me.
Knowing this, I offered to stage for free.
After my stage, I was hired by the exec.
On day 1, I told the exec that I want to one day run a kitchen and he seemed to take me under his wing.
When he has time, I find him teaching me things having to do with production, prep, scheduling staff, and even fixing/maintaining the equipment on the line. I do everything I can to show him that I am gratful for this.
He now needs a new KM/chef de cuisine.
I want this job.
I want to be responsible for the day to day success and effiency of this kitchen. And the experience of being a KM could really benefit my long term goals.
I want to apply for this and somehow show the exec that while I know I lack experience, I can make up for it with dedication, enthusiasm, time, detail, etc...
I want this, but I don't want to show disrespect to the chef by assuming that he would hire an noobie to run his kitchen or something...
QUESTION:
Should I bother? Can anyone see this happening?
IF SO:
How might I go about asking for a shot at this job?
I'm even willing to mention that with my lack of experience, I wouldn't expect to be paid the same as the last KM.

Am I trying to move up too fast?

Thanks for reading. I know it was a long post. Any advice would be greatly appreciated.