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We construct an analytically solvable simplified model that captures the essential features for primordial black hole (PBH) production in most models of single-field inflation. The construction makes use of the Wands duality between the constant-roll (or slow-roll) and the preceding ultra-slow-roll phases and can be realized by a simple inflaton potential of two joined parabolas. Within this framework, it is possible to formulate explicit inflationary scenarios consistent with the CMB observations and copious production of PBHs of arbitrary mass. We quantify the variability of the shape of the peak in the curvature power spectrum in different inflationary scenarios and discuss its implications for probing PBHs with scalar-induced gravitational wave backgrounds. We find that the COBE/Firas $μ$-distortion constraints exclude the production of PBHs heavier than $10^4 M_\odot$ in single-field inflation.