You will need a 12-quart stockpot for this recipe. We prefer andouille sausage here, but kielbasa can be used instead. Use baby potatoes measuring 1 to 2 inches in diameter. The boil spice is used in each component of the recipe, so be sure to make it first. Covering the pot after each ingredient is added helps the water return to a boil as quickly as possible. This recipe can be halved, but we recommend making the full amount of boil spice; use a large Dutch oven instead of a stockpot. We like to serve both sauces, but you can make just one and double it to have enough for serving.

FOR THE BOIL SPICE: Combine all ingredients in medium bowl. Set aside.

FOR THE BOIL: Combine 6 quarts water, boil spice, bay leaves, and quartered lemon in large stockpot. Cover and bring to boil over high heat. While water comes to boil, make sauces.

FOR THE RÉMOULADE: Whisk all ingredients in bowl until combined. Season with extra boil spice to taste and refrigerate until using.

FOR THE LEMON-GARLIC BUTTER SAUCE: Melt butter in small saucepan over medium-low heat. Stir in lemon zest and juice, boil spice, and garlic. Reduce heat to lowest possible setting and cook, stirring occasionally, for 10 minutes. Remove from heat and cover to keep warm.

When water comes to boil, add potatoes. Cover, just until water comes back to boil, then uncover. Adjust heat to maintain gentle boil and cook for 13 minutes. Add sausages and corn. Cover, just until water comes back to boil, then uncover and continue to cook at gentle boil for 7 minutes. Off heat, add shrimp; cover pot; and let stand until shrimp are just cooked through, about 4 minutes.

Using spider skimmer or slotted spoon, transfer potatoes, sausages, corn, and shrimp to large rimmed platter (discard cooking liquid). Transfer sausages to cutting board and slice on bias into thirds; return to platter. Sprinkle everything evenly with parsley and 1 tablespoon boil spice. Serve, passing rémoulade, butter sauce, lemon wedges, and remaining boil spice separately.