Feed most of the energy of star to lasers and end up with weapon that will melt planets across the galaxy.
Honestly, don't need to expand across the galaxy if have Dyson Sphere, which could be explanation of Fermi Paradox.
To minimize theta, we need to either increase D or decrease lamda.
Lets assume we would be able to make far infrared high power lasers, at say 10,000 nm = 1e-5 m wavelength.
Lets assume we would be able to make D, the diameter of our laser beam, similar to the diameter of a typical planet, for Earth it is ~13,000 km = 13e6 m.
Theta = 1e-5 / 3.14 / 13e6 ~= 1e-13 radians.
Sun is ~ 25,000 light years from the center of our galaxy, ~= 25e3 y 3e8 m/s 31.5e6 s/year ~= 1e20 m.
Laser beam diameter, there far away, would be: 1e-13 *1e20 = 1e7 m, similar to the diameter of Earth, not much further diverged, focused and delivering the wast amount of energy all over the planet thereby evaporating it to a gas.
Indeed, what you say about the melting far away planets is possible, in theory.