In a large non-stick skillet, heat 1 teaspoon of olive oil on medium-high heat. Add the onion and garlic, sautéing for 2-3 minutes, or until tender. Remove from heat and set aside. Place one of the buns in a food processor, and blend until small bread crumbs. Remove and set aside. Place mushrooms, lentils, 2 tsp mustard, thyme, pepper, and balsamic vinegar in a food processor, and pulse to combine. Combine the breadcrumbs, mushroom mixture, onion mixture, and the egg and stir until well blended. Using hands, make 4 equal patties out of the mixture. Put remaining olive oil in the skillet over medium-high heat. Add the 4 patties and cook for 4-5 minutes on each side, or until nicely browned. In the final minute of cooking, carefully add the goat cheese on top of burgers to melt. In a separate bowl, mix up the remaining mustard with mayonnaise. Spread mixture over bottom half of brioche bun, place patty on top, layer with arugula and finish with top half of brioche bun.