We present the first results from our survey of intervening and proximate Lyman limit systems (LLSs) at z~2.0-2.5 using the Wide Field Camera 3 on board the Hubble Space Telescope. The quasars in our sample are projected pairs with proper transverse separations R_{perp}_<=150kpc and line-of- sight velocity separations <~11000km/s. We construct a stacked ultraviolet (rest-frame wavelengths 700-2000{AA}) spectrum of pairs corrected for the intervening Lyman forest and Lyman continuum absorption. The observed spectral composite presents a moderate flux excess for the most prominent broad emission lines, a ~30% decrease in flux at {lambda}=800-900{AA} compared to a stack of brighter quasars not in pairs at similar redshifts, and lower values of the mean free path of the HI ionizing radiation for pairs ({lambda}_mfp_^912^=140.7+/-20.2h_70_^-1^Mpc) compared to single quasars ({lambda}_mfp_^912^=213.8+/-28h_70_^-1^Mpc) at the average redshift z~2.44. From the modeling of LLS absorption in these pairs, we find a higher (~20%) incidence of proximate LLSs with logN_HI_>=17.2 at {delta}v<5000km/s compared to single quasars (~6%). These two rates are different at the 5{sigma} level. Moreover, we find that optically thick absorbers are equally shared between foreground and background quasars. Based on these pieces of evidence, we conclude that there is a moderate excess of gas-absorbing Lyman continuum photons in our closely projected quasar pairs compared to single quasars. We argue that this gas arises mostly within large-scale structures or partially neutral regions inside the dark matter halos where these close pairs reside.