  We propose a method for calculating the nucleon form factors (FFs) of
$G$-parity-even operators by combining Chiral Effective Field Theory
($\chi$EFT) and dispersion analysis. The FFs are expressed as dispersive
integrals over the two-pion cut at $t > 4 M_\pi^2$. The spectral functions are
obtained from the elastic unitarity condition and expressed as products of the
complex $\pi\pi \rightarrow N\bar N$ partial-wave amplitudes and the timelike
pion FF. $\chi$EFT is used to calculate the ratio of the partial-wave
amplitudes and the pion FF, which is real and free of $\pi\pi$ rescattering in
the $t$-channel ($N/D$ method). The rescattering effects are then incorporated
by multiplying with the squared modulus of the empirical pion FF. The procedure
results in a marked improvement compared to conventional $\chi$EFT calculations
of the spectral functions. We apply the method to the nucleon scalar FF and
compute the scalar spectral function, the scalar radius, the $t$-dependent FF,
and the Cheng-Dashen discrepancy. Higher-order chiral corrections are estimated
through the $\pi N$ low-energy constants. Results are in excellent agreement
with dispersion-theoretical calculations. We elaborate several other
interesting aspects of our method. The results show proper scaling behavior in
the large-$N_c$ limit of QCD because the $\chi$EFT calculation includes $N$ and
$\Delta$ intermediate states. The squared modulus of the timelike pion FF
required by our method can be extracted from Lattice QCD calculations of vacuum
correlation functions of the operator at large Euclidean distances. Our method
can be applied to the nucleon FFs of other operators of interest, such as the
isovector-vector current, the energy-momentum tensor, and twist-2 QCD operators
(moments of generalized parton distributions).
