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Enormous Ly$α$ nebulae (ELANe) around quasars have provided unique insights into the formation of massive galaxies and their associations with super-massive black holes since their discovery. However, their detection remains highly limited. This paper introduces a systematic search for extended Ly$α$ emission around 8683 quasars at $z=$2.34-3.00 using a simple but very effective broad-band $gri$ selection based on the Third Public Data Release of the Hyper Suprime-Cam Subaru Strategic Program. Although the broad-band selection detects only bright Ly$α$ emission ($\gtrsim1\times10^{-17}$ erg~s$^{-1}$cm$^{-2}$arcsec$^{-2}$) compared with narrow-band imaging and integral field spectroscopy, we can apply this method to far more sources than such common approaches. We first generated continuum $g$-band images without contributions from Ly$α$ emission for host and satellite galaxies using $r$- and $i$-bands. Then, we established Ly$α$ maps by subtracting them from observed $g$-band images with Ly$α$ emissions. Consequently, we discovered extended Ly$α$ emission (with masked area $>40$ arcsec$^2$) for 7 and 32 out of 366 and 8317 quasars in the Deep and Ultra-deep (35 deg$^2$) and Wide (890 deg$^2$) layers, parts of which may be potential candidates of ELANe. However, none of them seem to be equivalent to the largest ELANe ever found. We detected higher fractions of quasars with large nebulae around more luminous or radio-loud quasars, supporting previous results. Future applications to the forthcoming big data from the Vera C. Rubin Observatory will help us detect more promising candidates. The source catalogue and obtained Ly$α$ properties for all the quasar targets are accessible as online material.