We present a new measurement of the Ly{alpha} forest power spectrum at 1.8<z<3.4 using 74 Keck/HIRES and VLT/UVES high-resolution, high-signal-to-noise-ratio quasar spectra. We developed a custom pipeline to measure the power spectrum and its uncertainty, which fully accounts for finite resolution and noise and corrects for the bias induced by masking missing data, damped Ly{alpha} absorption systems, and metal absorption lines. Our measurement results in unprecedented precision on the small-scale modes k>0.02s/km, inaccessible to previous SDSS/BOSS analyses. It is well known that these high-k modes are highly sensitive to the thermal state of the intergalactic medium, but contamination by narrow metal lines is a significant concern. We quantify the effect of metals on the small- scale power and find a modest effect on modes with k<0.1s/km. As a result, by masking metals and restricting to k<0.1s/km, their impact is completely mitigated. We present an end-to-end Bayesian forward-modeling framework whereby mock spectra with the same noise, resolution, and masking as our data are generated from Ly{alpha} forest simulations. These mock spectra are used to build a custom emulator, enabling us to interpolate between a sparse grid of models and perform Markov chain Monte Carlo fits. Our results agree well with BOSS on scales k<0.02s/km, where the measurements overlap. The combination of the percent-level low-k precision of BOSS with our 5%-15% high-k measurements results in a powerful new data set for precisely constraining the thermal history of the intergalactic medium, cosmological parameters, and the nature of dark matter.