Bettie is not a person in this instance; it's a super yummy, super sweet dessert. A member of the pie slash cobbler family, brown betties have a wonderful crumbling top on them. Often the topping is made up of lots of sugar, especially brown sugar. So, with something so sweet, you have to watch the wine pairing.

This article from the Stateman's Journal offers some wine suggestions along with some recipes for the cobbler/brown bettie fanatic:

Wine
One of the rules of pairing wines with dessert is that the wine always should be sweeter than the dish.
When selecting a wine to serve with the sugary Pear Cobbler or chocolatey Kentucky Derby pie, the options are limited to a late harvest Riesling, Ice Wine, Port or Moscato d'Asti.

Although the ice wine emphasized the pear flavor in the cobbler, the nutmeg was overamplified, while the dish accentuated the alcohol in the wine.

The late harvest Riesling and port fared even worse. The Moscato d'Asti, however, balanced the acids and sweetness well for both dishes.

The Blueberry Buckle paired harmoniously with the Moscato d'Asti as well. The higher acids, lemony flavors and spritz of the Moscato d'Asti accentuated the blueberry and cake flavors of the dessert.