It is shown that colloidal monolayers of dielectric microparticles with high refractive index (e.g., titania, zirconia) can convert incident, circularly polarized laser light into the lattice of photonic vortices that carry orbital angular momentum. Such particle monolayers are formed via self-assembly on various surfaces. Properties of the vortices are studied analytically, taking into account the symmetry of the problem. Vortex lattices of topological charges m=+-1 and two different polarizations are shown to be possible. Generation of the vortex lattices by the spherical and spheroidal particles irradiated by femtosecond laser pulses is studied using the finite difference time domain simulation. The vortex generation efficiency depending on the particle parameters is analyzed.