We developed a broadband two-layer anti-reflection (AR) coating for use on a sapphire half-wave plate (HWP) and an alumina infrared (IR) filter for the cosmic microwave background (CMB) polarimetry. Measuring the faint CMB B-mode signals requires maximizing the number of photons reaching the detectors and minimizing spurious polarization due to reflection with an off-axis incident angle. Sapphire and alumina have high refractive indices of 3.1 and are highly reflective without an AR coating. This paper presents the design, fabrication, quality control, and measured performance of an AR coating using thermally sprayed mullite and Duroid 5880LZ. This technology enables large optical elements with diameters of 600 mm. We also present a thermography-based nondestructive quality control technique, which is key to assuring good adhesion and preventing delamination when thermal cycling. We demonstrate the average reflectance of about 2.6% (0.9%) for two observing bands centered at 90/150 (220/280) GHz. At room temperature, the average transmittance of a 105 mm square test sample at 220/280 GHz is 83%, and it will increase to 90% at 100 K, attributed to reduced absorption losses. Therefore, our developed layering technique has proved effective for 220/280 GHz applications, particularly in addressing dielectric loss concerns. This AR coating technology has been deployed in the cryogenic HWP and IR filters of the Simons Array and the Simons observatory experiments and applies to future experiments such as CMB-S4.