We observed with ALMA three deeply buried nuclei in two galaxies, NGC 4418 and Arp 220, at $sim$0.2$$ resolution over a total bandwidth of 67 GHz in $f_{rm rest}$ = 215 - 697 GHz. Here we (1) introduce our program, (2) describe our data reduction method for wide-band, high-resolution imaging spectroscopy, (3) analyze in visibilities the compact nuclei with line forests, (4) develop a continuum-based estimation method of dust opacity and gas column density in heavily obscured nuclei, which uses the BGN (buried galactic nuclei) model and is sensitive to $log(N_{rm H_2}/{rm cm}^{-2}) sim $ 25 - 26 at $lambda sim 1$ mm, and (5) present the continuum data and diagnosis of our targets. The three continuum nuclei have major-axis FWHM of $sim$0.1$$-0.3$$ (20-140 pc) aligned to their rotating nuclear disks of molecular gas. However, each nucleus is described better with two or three concentric components than with a single Gaussian. The innermost cores have sizes of 0.05$$-0.10$$ (8-40 pc), peak brightness temperatures of ~100-500 K at 350 GHz, and more fractional flux at lower frequencies. The intermediate components correspond to the nuclear disks. They have axial ratios of $approx$0.5 and hence inclinations $stackrel{>}{sim} 60$ deg. The outermost elements include the bipolar outflow from Arp 220W. We estimate 1 mm dust opacity of $tau_{rm d,1mm} approx 2.2$, $1.2$, and $stackrel{<}{sim} 0.4$ respectively for NGC 4418, Arp 220W, and Arp 220E. The first two correspond to $log(N_{rm H}/{rm cm}^{-2}) sim 26$ for conventional dust-opacity laws, and hence the nuclei are highly Compton thick.
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