I would run it under cool water, or soak it in a container of cool water for a few minutes in the morning before daycare.
I also agree with PPs that some frozen chunks are totally fine, as long as it is broken up enough to fit into the bottle and so you can swirl enough to get the fat off the walls of the bag
I put it in the fridge the evening before. By morning it's usually 50% defrosted, but the ice is easy to break while its still in the bag. I break the chunks so they're small enough to get in the bottle, and send it to daycare like that. They warm bottles anyways, so it's not like she's drinking icey milk
Re: Defrosting properly
I would run it under cool water, or soak it in a container of cool water for a few minutes in the morning before daycare.
I also agree with PPs that some frozen chunks are totally fine, as long as it is broken up enough to fit into the bottle and so you can swirl enough to get the fat off the walls of the bag