I am embarking on my second hand quilted piece, a small lap quilt/twin topper. The last quilt I hand stitched was mostly done on a frame but I ended up with it on my lap and I abandoned any and all frames/hoops. So, I started this one right off the bat, on my lap. I know the answer is 'whatever you find the most comfortable' but I want this quilt to be and look exceptional as it will be included in a silent auction.What method will give me this?

I think you use what is comfortable and gives you the best stitches. I bought a very fancy frame and used it only once. I found my stitches look better when the quilt is in my lap.

I prefer to use just a lap hoop. It gives me just enough tension to work with.

The same goes for me, but I'm not an expert yet, lol!!

Yes, I found that I was more comfortable with a bit of loose tension. I,too, bought a quilting frame but I had to be a contortionist in order to get a real feel for the tension and to reach sections. I can manipulate the quilt better when on my lap.
As an aside, I like the look of a hand stitched quilt. Quilts sewn with long arm/computerized or whatever they are machines are so over worked with patterns etc that IMHO it takes away from the fabric and the piecing. There's a place for these but that's a lot of what I see in the Picture thread on this site.