To make the flax egg, combine the flax meal and water in a small bowl and stir. Set aside.
In a large saucepan, heat 1 tablespoon of olive oil over medium heat.
Once the oil begins to shimmer, add the mushrooms and cook until most of the juices have evaporated, 7-8 minutes.
Add the soy sauce and paprika and cook for 2-3 more minutes. Set aside in a large bowl.
Heat another tablespoon of olive oil over medium heat. Once the oil begins to shimmer, add the onion and cook for 3-5 minutes, until semi-translucent. Add the garlic and cook for 2 minutes. Add the salt and pepper and cook for another 2 minutes.
Transfer the onion to a food processor, then add the lentils, oats, and flax egg. Pulse until chunky, about 10-15 pulses.
Transfer the lentil mixture to the bowl with the mushrooms. Mix well, using your hands if needed, then shape into 5 patties. Place on a baking sheet and cover with plastic wrap, then freeze for 20 minutes.
While the patties are freezing, prepare the fries. In a medium bowl, toss the potatoes, a drizzle of olive oil, the paprika, garlic powder, salt, and pepper until well-coated.
Spread the fries on a baking sheet and bake for 15 minutes.
Remove the patties from freezer. In a large saucepan, heat the remaining tablespoon of olive oil over medium heat. Once the oil begins to shimmer, add 2-3 patties, careful not to overcrowd the pan. Cook for 5 minutes, then flip and cook for 5 minutes on other side. Repeat with the remaining patties.
To assemble the burgers, top each bun with a patty, then layer on dijon mustard, tomato slices, and arugula.