Pre-heat BBQ to high heat. Place the Toulouse sausages on the grill and cook for 8 minutes, flip and cook on the other side for another 5-6 minutes. Remove from heat and allow to rest for 10 minutes. Slice into 1/2-inch rounds.
Cut the pizza dough ball into 2 equal pieces. Lightly flour the countertop and flatten one of the dough balls by hand. Use a rolling pin to roll the dough into a 12-16 inch round, rotating and stretching the dough as you go. Transfer the dough to a lightly floured pizza peel (or rimless baking sheet) and stretch to re-shape if necessary.
Oil the BBQ grill grates and decrease the heat to medium.
Brush the dough with 1 to 2 teaspoons of olive oil and flip onto the hot grill, oiled side down. Close the lid and cook until the bottom of crust is golden brown, approximately one to two minutes.
Brush the raw side of the dough with 1 to 2 teaspoons olive oil, then immediately flip using the peel (or rimless baking sheet). Spread three tablespoons of artichoke tapenade onto the dough, then top with grated feta and mozzarella, sliced Toulouse sausage and torn basil.
Close the lid and cook until the bottom of crust is golden browned and cheese has melted, about two to four minutes.
Using the peel (or rimless baking sheet), remove the pizza to a cooling rack and let rest for 3 minutes. Top with baby arugula, drizzle with two tablespoon olive oil. Slice into desired size.
Place the artichoke hearts, garlic, cayenne powder, olives, capers, oil, lemon juice and sea salt in a food processor and puree until smooth. Add the cheese and puree for another 15 seconds. Set aside.
Store extra artichoke tapenade in a resealable container in the fridge for up to a month.