Dyson Sphere: The Empire has two, although neither by that name, the concept *there* having been devised by Cirys Minaxianos. One, a Cirys bubble, is at Esilmúr (Imperial Core); a thin layer of light-sail material surrounds the star, held up by photon and solar wind pressure in much the same manner as a statite, and managed by magnetic manipulation and selective releases of solar wind. The material, in turn, is coated with energy-producing solar foil, powering massive arrays of antimatter production facilities.
The other, a three-layered incomplete Cirys swarm is at Corícal Ailék, where the Transcend keeps its brain: the innermost being golden-winged energy-collection complexes skimming the corona of the star and powering the rest of the swarm, the middle layer being fractal chandeliers of organic crystal nanocircuitry, providing the Transcend with its massive processing power, and the outermost being the hexterranes, a lattice of hexagonal habitat-plates providing living space here at the heart of the light.
Big Dumb Object: Oh, plenty. Leaving aside those belonging to elder races – and thus little known due to the ability to enforce the Do Not Taunt rule – the Empire has a partially completed Dyson Sphere in the works at Corícal Ailek (it’s where the Transcend keeps its brain; and it’s the swarm kind, not the shell kind) and another at Esilmúr (a primary antimatter production facility; the solar-wind-inflated-bubble kind), and the Photonic Network has at least one under construction somewhere in its interior. Then there are assorted Stellar Husbandry Arrays floating over Imperial stars and keeping them running nice and smoothly and for rather longer, in theory, than they otherwise would.
Some spacedocks/construction slips probably qualify in the minor leagues due to the sheer size of a fleet carrier/grapeship megafreighter, as does the Conclave Drift drift-habitat that houses the Conclave of Galactic Polities, and I suspect that the manufacturer would strongly contend that the Ring Dynamics, ICC Interstellar Stargate (Mark III), a.k.a. the big framework/space station/support machine that wormhole ends get wrapped in, absolutely counts for these purposes.
And there’s more…
Abusing the Kardashev Scale For Fun and Profit: Working their way up to this right now, the Empire has a couple of Dyson spheres, one under construction, the other completely dedicated to energy generation, plus other energy-generation facilities scattered here and there and about the place. This makes them, I believe, a solid Type II, although given the constraints of light speed, it’s going to take them a while to work up to III.