You can't plow very close together in time. Soil needs to be packed for a plow to work. Besides, it defeats the purpose of burying sod, only to later roll it back to the top. Plowing in the fall works well, as all the fodder and sod gets buried and will rot by spring. You will have a nice clean garden for planting.... until the weeds naturally start to grow. Sorry man, weeds are God's curse to us. It's in the Bible.
Keep it mowed down the rest of the summer and plow it this fall, later the better. Like after the leaves fall and grass dies off. October/November. Being you are up north, maybe late September/October. If you have a wet fall, you can plow in the spring. Couple years ago we had a wet fall, but mild winter. I plowed in February, and tilled and planted in April. Plowing works well when the soil is damp and even heavy. If you catch it right, you will have to let it dry a couple days before you till. Not mud, mind you. Just "heavy".