Here are two totally different New Year traditions. I take that back. They share a round shape but that is where I draw the line!
The first one is the New Year Day cookie (Portzelky). It is a drop donut of sorts with raisins that I had growing up.
1 envelope yeast
1/4 cup warm water
5 1/2 cup AP flour
1 1/2 t sugar
1 1/2 t salt
pinch nutmeg
1 1/2 cup warm milk
3 eggs
1 1/2 cup raisin, currants, etc (plump the raisins in water first - blot dry well to reduce oil splattering)
Dissolve yeast in the water, let proof.
Mix flour, sugar, salt, and nutmeg together in large bowl.
Lightly beat eggs with the milk.
Add the yeast mixture to the egg/milk mixture and then pour that into the flour mixture.
Mix well
Stir in raisins.
Let sit until raised like a poolish (good and bubbly). Right before you begin to fry, tap the bowl on the counter or push down the dough a bit.
Drop by cookie scoop into hot oil. 
Remove from oil and drain on paper towels. Once drained roll in sugar and place on a wire rack to dry so they do not become soggy.
Some people cover them with glaze, others with sugar (powdered or fine). I usually go with organic sugar because I like look of the grains and they don't "melt" and make the fritters soggy.
I make mine throughout the year play around with what ever flavors and additives I have around. Craisins and orange zest was a favorite. Sometimes I add cinnamon or nutmeg to the sugar before I roll them in it.
Now for the unfamiliar tradition. Mochi!
From what I understand, these are a Japanese New Year custom. Since I don't have a steam kneader I opted to try the microwave rice flour version.
Microwave Chocolate Mochi (courtesy of Eleanor Urakawa and various tips I found on the web)
1-1/2 cups mochiko (rice flour)
1-1/2 cups water
1/2 cup sugar
Pinch salt
1/2 cup chocolate chips
Katakuriko (potato starch) or kinako (roasted soybean flour), for dusting
Filling
Mix mochiko, water, sugar, chocolate, and salt in a bowl. Lightly coat a microwavable bowl with cooking spray. Pour mixture into the bowl and cover with plastic wrap or a microwave cover. Microwave 3 minutes on low, then 3 minutes on medium and 3 minutes on high, for a total of 9 minutes. This varies depending on your microwave. Cook until it is almost translucent, stirring occasionally.
I choose two different fillings; nutella and a raspberry marzipan mixture.
Look at the beautiful new jar of nutella! It is about to be marred for life LOL!
Let the mixture cool a bit so that you don't burn yourself. Many sites said to roll it in a tube or flat rectangle and then cut. I found it best to just use a wet cookie scoop.If you have silpat, I would strongly recommend it.
Dust the mat with corn starch or potato starch and plop down a scoop of the mixture.
Dip your fingers in the starch. Quickly but gently work the blob into a disk. Be careful not to drag your hands but to pat the dough instead. Don't let them rest on the dough or they will stick. If your hands feel sticky, wash and then re-coat with starch or the sticky on your fingers will stick to the disk and you will literally have a mess on your hands.I read of sticky horror stories on the web but honestly did not have any trouble with these. But then again, dough and I usually get along :)
Once you have a disk, place in your filling, and then seal like a beggars purse.
Dust off the excess and gently turn the mochi so that the seam is on the bottom. You can choose to put them on a starched plate or in cupcake liners.
Let firm. They can be served room temp, chilled, or even frozen. The Mochi was a fun little project. I am overall happy with they way my first try turned out. Next time I will choose a firmer filling that I can roll in a ball so they aren't so flat and there is also more filling. I also want to try the ice cream version.
