Preheat the oven to 400 degrees F. Preheat a grill.
Rub the salmon fillets with oil and thyme. Season with salt and pepper. Grill the salmon until marked on both sides and medium doneness, about 2 to 3 minutes per side. Remove from the grill and set aside.
Cut the bread into 8 (1/2-inch thick) slices. Place the bread on a sheet pan and brush with olive oil. Place the bread in the oven and cook until lightly toasted, about 10 minutes.
Remove the ends from the cucumber and slice in half horizontally. Slice on a mandoline into 1/8-inch thick slices.
Spread the Salmon Remoulade on 2 slices of the toasted bread. Place the salmon on 1 slice of the bread. Top with a few slices of the cucumber and 2 slices of tomato. Toss the arugula with some of the House Dressing in a small bowl. Top the tomatoes with 1/4 of the arugula. Top with the other slice of bread and cut in half diagonally. Repeat with the remaining ingredients.
Place the lemon aioli in a food processor. Add capers, mustards, ketchup, tarragon, red onion, dill, chives, basil, and smoked salmon and pulse until well combined. Taste for seasoning.
In a food processor, combine the egg yolk, mustard, and roasted garlic and process. Add the lemon juice and pulse to combine. With the motor running, slowly add the oil until it is all added and the mixture is emulsified. Add the lemon zest and thyme and pulse to combine. Season with salt and pepper.
In a small bowl, whisk together the shallots and the mustard. Whisk in the vinegars and then the olive and vegetable oil. Season with salt and pepper, to taste. Transfer to a covered container and refrigerate until needed.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.