Hope this helps some.
Attaching shaker drawer fronts.
You could also run your drawer handle through the panel and use those screws to help hold the front on with some screws on the side into the stiles.
Take the drawer and drill a slight hole on the 4 corners of the front inside of the drawer leave at least 5 1 space from the ends this is where the screws will be placed to add the front.
Then you d have more thickness to screw into.
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I think a better alternative if you re set on this design is to use a 1 2 panel.
Materials needed for drawer fronts 3 4 thick pine board strips 2 1 4 wide 3 8 thick plywood 1 1 4 pocket hole screws finishing nails construction adhesive paint drawer handle.
Make sure the slight hole is only deep enough for the screws head.