Set out six 5-inch-wide, 6-ounce ceramic pie dishes. Fill a large bowl three-quarters-full with cold water, and place it near the stove. (This cold-water bath will be used to stop the caramel from cooking.) Combine 1 1/4 cups sugar and 1/2 cup cool water in a small, heavy-bottomed saucepan over medium-high heat. Cook, stirring, until the sugar dissolves. Boil, brushing the sides occasionally with a wet pastry brush, until syrup turns deep amber (330 to 350 degrees on a candy thermometer). Don't stir the sugar once it's boiling; this only encourages recrystallization, which occurs when the sugar does not dissolve evenly. Quickly dunk bottom of pan in cold water, then remove from bowl.
Quickly divide caramel among pie dishes; swirl to coat the bottoms. Set dishes aside to cool, about 15 minutes. (This can be done several hours ahead of time; the dishes can be left at room temperature.)
Heat oven to 325 degrees. Pour milk into saucepan over high heat. Bring just to a boil; set aside. In a bowl, combine pumpkin puree, remaining 3/4 cup sugar, ginger, spices, salt, eggs, and egg yolks. Mix in vanilla and milk; pass through fine sieve, discarding solids. Divide mixture between the dishes. Transfer to a rimmed baking sheet large enough to hold them.
Place baking sheet on center rack of oven; loosely drape a piece of foil over top of baking sheet (do not crimp or seal edges). Lift a corner of the foil and pour very hot water into sheet pan to reach 1/2 inch up the sides of the pie dishes. Bake 60 to 65 minutes, or until center of flan is nearly set; a thin-bladed knife inserted into center should come out clean. Remove the dishes from the water bath, dry underside and let slightly cool on wire rack. Refrigerate to set, several hours or overnight. (Let the water bath cool completely before removing it from the oven.)
To unmold flan, run a knife carefully around edges, without cutting into flan. Place plate over top of pie pan and invert. Flan should fall with caramel sauce surrounding the flan. Serve with whipped cream. Garnish with cinnamon sticks.