Stuffing: Rough chop shrimp, taking care to remove any debris. Briefly saute in oil. Add tarragon, salt, and pepper. Deglaze with stock, but do not reduce. Remove from heat and add goat cheese to bind. Stir in goat cheese off the heat so it gently melts. Transfer to a container and refrigerate.
Dough: In bowl of heavy-duty mixer, stir together water and yeast until yeast granules dissolve. Whisk in 1 cup flour to make thick batter. Cover bowl loosely with plastic wrap. Let stand for 30 minutes, or until sponge has doubled in volume and is bubbly.
In mixing bowl, whisk together milk, butter, and eggs. In separate bowl, whisk together 6 1/2 cups of remaining flour and salt. Lightly oil large bowl.
In bowl of heavy-duty mixer, using paddle attachment beat sponge on low speed until smooth. Add milk mixture and beat until blended. Using rubber spatula, scrape down sides of bowl. Add flour mixture one cup at a time. Increase speed to medium and beat dough for 5 minutes, or until is comes away from side of bowl and wraps around paddle. If dough is sticking to side of bowl at this point, add remaining 1/2 cup flour, 1 tablespoon at a time until dough stops sticking.
Scrape dough into a lightly oiled bowl (it will be sticky). Cover bowl with plastic wrap and let dough rise for 45 to 60 minutes or until it has increased in volume 1 1/2 times.
Lightly dust large baking sheet with flour. Dust work surface with flour. Scrape dough onto work surface and lightly sprinkle top of dough with more flour. Dust hands with flour. Gently press down on the dough to deflate it. Flatten dough into circle about 1/4-inch thick.
Onto floured surface, roll dough out firmly and smoothly to 3/4-inch thickness. Using a sharp blade, square off dough as evenly as possible. Lightly brush entire surface with egg whites and mark 1/2 of prepared dough with a floured 3-inch cutter. Do not cut through dough. Place heaping tablespoon of shrimp mixture in center of marked circles. Be sure to keep away from outside ring.
Gently fold remaining 1/2 of dough back over marked circles. Using fingers, press out air pockets from around shrimp filling. Cut through dough using cutter centered around the filling. Save dough scraps and roll out for more beignets. Deep fry at 375 degrees F for approximately two minutes on each side, or until golden brown.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.