Place the chocolate and the sweetened condensed milk in a medium heavy-bottomed sauce pot (I use a 3-quart pot). Heat over low heat, stirring constantly, until the chocolate has melted and the mixture is smooth.
Turn off the heat and stir in the vanilla extract.
Transfer the chocolate mixture to a large bowl (one that will be large enough to fold in the whipped cream later); allow to cool completely, uncovered, about 1½ hours. Note: Don't cover the bowl, as you don't want any condensation to build up and get in your chocolate.
Put the bowl that you will whip the cream in in the refrigerator to chill while the chocolate cools (having the heavy cream and the bowl super cold is key to ensuring that your cream whips up easily).
Once the chocolate mixture has cooled completely, beat it with a wooden spoon until smooth.
In a large bowl, beat the heavy whipping cream with a hand-held or stand mixer (or by hand with a whisk) until stiff peaks form. Start on a low speed until the cream starts to thicken up (this will prevent the cream from splattering all over). Then increase the speed to medium until you get to stiff peaks (when you lift out the beaters, the peaks should stay firm and not fall over on themselves). Note: Be very careful not to whip the cream too much, or it can start to become grainy (and you will eventually end up with butter)! Take half of the whipped cream and gently fold it into the chocolate mixture until it is mostly combined (it is okay, at this point, if the whipped cream isn't totally incorporated; we're just trying to loosen up the chocolate). Be very gentle when folding...you don't want to deflate all that air that was whipped into the cream.To fold: take your spatula or spoon and gently go into the middle of the mixture (be sure to get all the way down to the bottom of the bowl), scooping up the mixture from the bottom and folding it over on itself. Turn the bowl a quarter turn and keep repeating until the mixture is well-combined.
Gently fold the remaining whipped cream into the chocolate mixture until it is completely incorporated and the mixture is one even color with no streaks of whipped cream or chocolate.
Spoon the mousse into 8 dessert cups/dishes. Cover and refrigerate until thoroughly chilled, at least 3 hours.
Top with whipped cream and shaved chocolate, if desired.