Heat the olive oil in a large stock pot and add the onions. Cook over medium-high heat, stirring often, until the onions start to soften, about 8 minutes.
Add a few tablespoons of water to deglaze and continue cooking until they are brown and caramelized, about 5 minutes.
Add the mushrooms, paprika, dill, salt, and pepper. Lower the heat slightly, cover, and cook 5 minutes, stirring frequently.
Add the soy sauce and vegetable stock. Uncover and simmer about 10 minutes.
In a separate pan, mix the olive oil and flour to form a thick paste. Cook over medium heat, stirring constantly, for 1 minute. Whisk in the dairy-free milk and cook until steaming and slightly thickened.
Pour the dairy-free milk mixture into the soup. Stir in the lemon juice just before serving.