EAGLE has no say over the outcome of the thickness of the actual copper layers themselves, since you'll be etching the board yourself and you're the one picking the copper board stock. However, you can increase the size of the annular ring to give yourself more room to drill a hole close to the center of a PTH.
Using the Info tool (the button that is the letter i), click on the PTH you are interested in and adjust the Diameter from auto to another value that is greater than your Drill size. Click Apply and visually inspect the hole - if it's still not as much copper as you want, increase the Diameter again until it's as thick as you want.
If you want to do this to all your PTHs, just use the Change tool, go to Diameter, and click the ... option at thebottom to set a custom value. This would ideally be the Diameter you chose when playing around with a single PTH. Then just start clicking on the other PTHs you want to update them.
This will only work, however, on PTHs you defined manually on the board layout. If some PTHs are part of the part's package, for example, you will have to edit their Diameter manually in the package itself, not in the board layout.
Best Answer
There are primarily two reasons to use teardrops:
That being said, in professionally made PCB's teardrops are rarely needed. It's almost more of an aesthetic thing than a solution to a real problem. I've done many boards with and without teardrops and I have yet to notice a difference. In my opinion, they are more trouble than they are worth.