Answer: The answer to your question depends somewhat on why you are using the mulch. If you are going to plant a shrub and mulch around it for example, I would suggest loosening the soil and buring the sod upside down at the bottom of the planting hole, then plant and then mulch over the bare soil at the root area.
If you are planting a lot of plants in one area, it can be easier to remove the sod and till the whole bed as part of the preparation rather than dig a multitude of individual holes close together. Most tillers have difficulty tilling up sod, so it can be removed and composted or replanted elsewhere. Then you would mulch the overall area.
If you are simply wanting to kill off lawn grass to create a mulched area (or mulch around an existing shrub for example), you can mow the lawn short and cover it with several layers of damp newspaper or cardboard and then place the mulch in a deep layer over top of that. This should smother the lawn by excluding light. This will eventually make a nice planting area, or you can just continue mulching it. The paper or cardboard will rot away eventually.
Some people have successfully created planting areas by smothering the lawn with paper and mulch and then planting through that. In my experience this can work, especially for a shrub bed, but it does not work that well for perennials or vegetables. Especially on a heavy clay soil the mulch tends to stay as a distinct layer on top of the clay, rather than working its way downward into the soil as it can on sandy loam. If this is what you have in mind, I would suggest you experiment with a smaller area and see how it does for you before you try a large area.
I hope this helps.
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