Just a few thoughts. The answer to your question depends on a number of key factors. To focus, let us consider a nonlinear term like $F(D_x^k u)$ and let us work in one space dimension.

1) How large is $k$? If $k\le2$ you can linearize the equation and work in Sobolev spaces. Of course you need some structural assumption on the nonlinearity (otherwise you may take e.g. $F(u_{xx})=\pm 2i u_{xx}+...$, and create all sorts of difficulties). There are some classical works by Kenig, Ponce, Vega on this (see "The Cauchy problem for the quasilinear S.e." around 2002 I think) which more or less give the complete picture from the classical point of view i.e. without trying to push below critical Sobolev etc. So if this is the case what are you exactly looking for? if you prefer to work in smaller spaces, what you need is a 'regularity' result, i.e., if the data are in some smaller space, this additional regularity propagates and the solution stays in the same space for some time. There are some results of this type, in classes of analytical or Gevrey functions; but see below.

2) If $k\ge3$, then the same remark as in (1) applies, you need some strong structural assumptions on the nonlinearity. Indeed, now the poor $u_{xx}$ is no longer the leading term and the character of the (linearized) equation is entirely determined by the Taylor coefficient of $D^ku$ in the expansion of $F$. So then a careful case-by-case discussion is necessary. Unless...

3) unless, and we come maybe closer to your question, you decide to work in MUCH smaller spaces than $C^\infty$. Gevrey classes are roughly speaking classes of smooth functions such that the derivatives of order $j$ grow at most like $j!^s$. For $s=1$ you get analytic functions. For $ 1 < s < \infty $ you get larger classes $G^s$, with quite nice properties (to mention just one, you have compactly supported functions in these classes). For $ 0 < s < 1 $ the classes $G^s$ are rather small, strictly contained in the space of analytic functions. The only reason why $G^s$ for $s<1$ are useful is that you can prove a sort of very general Cauchy-Kowalewski theorem, local existence in Gevrey classes, for any evolution equation $u_t=F(D^k_xu)$, provided $s<1/k$. No structure is required on $F$, only smoothness. Contrary to the appearance this is a weak result, e.g. you can solve locally both $u_t=\Delta u$ and $u_t=-\Delta u$ in $G^{1/2}$ (and globally in $G^s$ for $s<1/2$ ), so you are essentially trivializing the equation and forgetting all of its structure. But if this is what you need I can give you pointers.

EDIT (since this does not fit in the comments):
I am a bit rusty on these topics, now I start to remember more details.

1) $s<1$. There is a problem with working in $G^s$ for $s<1$, and it is that this space is unstable for product of functions. So also in this case you need very special nonlinearities to work with. But the linear theory is straightforward and you can find an account maybe here. Also Beals in some old paper developed semigroup theory in Gevrey classes. BTW, if you find a way to handle products this might be quite interesting.

2) $s>1$. The Mizohata school and other japanese mathematicians worked on NLS in Gevrey classes quite a lot, see e.g. this paper

3) The Nash-Moser approach might be useful provided you can prove that the linearized of your operator is solvable in every Sobolev class, with a fixed loss of derivatives. If you want to try this route, the best introduction to the theory I know of is in Hamilton's 1992 (?) Bull. AMS paper. It's very long but extremely readable, give it a try.