Oscillatory integral giving me the willies
Clearly the integral as stated diverges, so one needs to regularize it. To that end, consider $t$ complex with small negative imaginary part, to ensure that it converges at $E\to\infty$.
The integral directly matches the following integral representation of the Bessel function of the second kind: $$ K_{\nu }(z)=\frac{\sqrt{\pi } z^{\nu } }{2^{\nu } \, \Gamma \left(\nu +\frac{1}{2}\right)} \, \int_1^{\infty } \left(t^2-1\right)^{\nu -\frac{1}{2}} e^{-z t} \, \mathrm{d}t $$ valid for $\mathfrak{Re}(\nu) > -\frac{1}{2}$ and $\mathfrak{Re}(z) > 0$.
Thus: $$ \frac{1}{4 \pi^2} \int_m^\infty \mathrm{e}^{-i t \mathcal{E}} \sqrt{ \mathcal{E}^2-m^2} \mathrm{d} \mathcal{E} = -i \frac{m}{4 \pi^2 t} K_1\left( i m t \right) $$ By means of regularization we proclaim that integral equal to the rhs even for real $t$. Expanding: $$ -i \frac{m}{4 \pi^2 t} K_1\left( i m t \right) = -i \frac{m}{8 \pi t} H_1(m t) + i \frac{m}{4 \pi t} \operatorname{sign}(t) J_1(m t) $$ This allows to conclude that for $t \to +\infty$, the expression is proportional to $\mathrm{e}^{-i m t} \frac{\mathrm{e}^{-i 3 \pi/4}}{8} \sqrt{m} (\pi t)^{-3/2}$
Considering the integrand as the Fourier transform of a tempered distribution, it makes sense then to write
$$
\int_{m}^{+\infty}\sqrt{E^2-m^2}e^{-iEt}dE = -\frac{\partial^2}{\partial t^2}\int_{1}^{+\infty}\frac{\sqrt{\rho^2-1}}{\rho^2}e^{-imt\rho}d\rho.
$$
Now, in the complex plane, let us consider the contour in the figure: a quarter of circle centered at $1$ of radius $R$ with arc going from $R$ to $1-iR$ and a small indent around $1$. Integrating
$$
f(z)=\frac{\sqrt{z^2-1}}{z^2}e^{-imtz}
$$
along such a contour and choosing the branch cut from $-1$ to $+1$, we get a vanishing contribution (by Jordan's lemma) from the arc and hence
$$
\int_{1}^{+\infty}\frac{\sqrt{\rho^2-1}}{\rho^2}e^{-imt\rho}d\rho =
\int_0^{+\infty}\frac{\sqrt{y^2+i2y}}{(1-iy)^2}e^{-imt(1-iy)}dy.
$$
Differentiating twice, by the above consideration,
\begin{align}
\int_{m}^{+\infty}\sqrt{E^2-m^2}e^{-iEt}dE
&=m^2\int_0^{+\infty}\sqrt{y^2+i2y} \, e^{-mt(y+i)}dy
\end{align}
and rescaling $s=mty$
\begin{align}
\int_{m}^{+\infty}\sqrt{E^2-m^2}e^{-iEt}dE
&= e^{-imt}\sqrt{m}t^{-3/2} \int_0^{+\infty} \sqrt{\frac{s^2}{mt}+i2s}\,e^{-s}ds\\
&= e^{-imt}\sqrt{m}t^{-3/2} \left( \sqrt\frac{i\pi}{2} + O\left(t^{-1}\right) \right),
\end{align}
asymptotically as $t\to\infty$.