Current methods for fabricating microparticles offer limited control over size and shape. Here, we demonstrate a droplet microfluidic method to form polyhedral microparticles with controlled concavity. By manipulating Laplace pressure, buoyancy, and particle rheology, we generate microparticles with diverse shapes and curvatures. Additionally, we demonstrate the particles provide increased capture efficiency when used for particle-templated emulsification. Our approach enables microparticles with enhanced chemical and biological functionality.