Heat vegetable oil in large pot over medium heat; cook and stir onions, carrots, and celeriac until beginning to soften, about 5 minutes. Stir in potatoes and vegetable stock. Simmer until potatoes are tender, 15 to 20 minutes. Add parsley and blend potato mixture with a stick blender until soup is smooth. Stir half-and-half, salt, black pepper, and cumin into soup.
Ladle soup in bowls and garnish with tomatoes and hot dogs.