We observed with Atacama Large Millimeter/submillimeter Array three deeply buried nuclei in two galaxies, NGC 4418 and Arp 220, at ∼0.″2 resolution over a total bandwidth of 67 GHz in f 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 buried galactic nuclei (BGN) model and is sensitive to ∼ 25–26 at λ ∼ 1 mm, and (5) present the continuum data and diagnosis of our targets. The three continuum nuclei have major-axis FWHMs of ∼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 ≈0.5 and hence inclinations ≳60°. The outermost elements include the bipolar outflow from Arp 220W. We estimate 1 mm dust opacity of τ d,1 mm ≈ 2.2, 1.2, and ≲0.4, respectively, for NGC 4418, Arp 220W, and Arp 220E. The first two correspond to for conventional dust-opacity laws, and hence the nuclei are highly Compton thick.