Let's begin by pointing out the following: you will not find monotone examples for the simple reason that a nontrivial such deformations creates a class of nonzero symplectic area, while the Chern class is always vanishing. The best you could hope for is Calabi-Yau, and such examples indeed exist.

Now general observation: In the case when $T^*N \setminus N$ has no second cohomology with $\mathbb{R}$-coefficients, e.g. if $N=S^2,$ then for small closed forms $\sigma$ one can use Moser's trick to show that any compact Lagrangian submanifold of $T^*N \setminus N$ is preserved (up to smooth isotopy) after turning on a sufficiently small magnetic potential.

A more concrete example: taking $\sigma$ to be the area form on $S^2,$ we obtain the total space of the line bundle $\mathcal{O}(-2)$ on $\mathbb{C}P^2$ with its standard Kähler form. (The first reference coming to my mind is 2.4A in [Y. Eliashberg and L. Polterovich; Unknottedness of Lagrangian surfaces in symplectic 4-manifolds] but maybe there is something more to the point). Unlike $T^*S^2$, the latter symplectic manifold is an open toric Calabi-Yau manifold. Note that its Fukaya category is nontrivial by the proof of Theorem 2.2 from [Fuakay-Ohta-Ono-Oh; Toric degeneration and non-displaceable Lagrangian tori in $S^2 \times S^2$], since it has a balanced toric fibre (again, there is probably a more direct reference). Finally, note that the latter fibre corresponds to the Albers-Frauenfelder torus inside $T^*S^2,$ and can be obtained by an application of Moser's trick as above.