Proof the yeast: In a small bowl, combine warm water (100F for fresh yeast, 110F for dry yeast), 1/2 tsp sugar, and yeast. Set aside in a draft-free spot for 10 to 20 minutes. If mixture is not full of bubbles by this time, the yeast is too old. Place 1/3 c flour and 1 egg in food processor with dough blade. Process a few seconds until mixed. Add the yeast mixture and stir with a rubber scraper until smooth. Sprinkle the remaining flour over the mixture but do not mix it in. Cover and let stand for 1 1/2 to 2 hours.
Kneading the Dough: Add the remaining sugar, salt, and 2 cold eggs. Process 1 1/2 minutes or until the dough is smooth, shiny, and cleans the bowl. Let rest 5 minutes with the feed tube open. Add the butter in 2 batches and process for 20 seconds after each addition or until incorporated. (The butter must be soft so as not to over-tax the motor. If the processor should stall, let it rest 5 minutes.)
First Rise: Scrape the dough into a lightly buttered bowl. It will be very soft and elastic. Sprinkle lightly with flour to prevent a crust from forming. Cover the bowl tightly with plastic wrap and let rise in a warm place (80F, but not above or dough will taste sour) until double in bulk, about 2 hours. Refrigerate for at least 30 minutes to 1 hour. Deflate dough by gently stirring it and refrigerate for another hour.
Redistributing the Yeast: Turn dough onto a lightly floured surface and gently press it into a rectangle. Fold into thirds like a business letter and again press out into a rectangle, lightly flouring the surface as needed to prevent stickiness. Fold it again into thirds and dust it lightly with flour on all sides. Wrap it loosely but securely in plastic wrap and then foil. Refrigerate for 6 hours to 2 days to allow dough to ripen and harden. Gently deflate the dough by kneading lightly with floured hands, and press into prepared pan. Cover the dough loosely with buttered plastic wrap and let rise in a warm, preferably humid place away from drafts for 1 1/2 to 3 hours or until it has almost tripled in bulk. It will reach the top of a 2-inch cake pan. Place a baking sheet in the oven and preheat to 425. The hot baking sheet will boost the "oven spring" -- the sudden expansion of the dough during the first few minutes of baking. Place the cake pan on the hot baking sheet and bake for 5 minutes. Lower the temperature to 375 and bake for 20 minutes or until a skewer comes out clean. Unmold the brioche and cool on a rack. When ready to complete the cake, trim the top, bottom, and sides with a serrated knife so that the brioche measures 7 inches by 1 1/2 to 1 3/4 inches. Place in a 9-inch pan and pour the syrup on top. Let stand for 10 minutes or until the syrup is absorbed, turning cake over as necessary to aid absorption.
Making the Syrup: In a small saucepan with a tight-fitting lid combine the sugar and water and bring to a rolling boil, stirring constantly. Cover immediately, remove from heat, and cool completely. Transfer to a liquid measuring cup and stir in the rum. Add water as needed to total 1 3/4 c syrup.