In a large saucepan, cover the lentils with the water and bring to a boil.
Meanwhile, in a medium soup pot, heat 1/4 cup of the olive oil until shimmering. Add the onions, carrots, celery and sliced garlic. Cover and cook the vegetables over moderately high heat, stirring occasionally, until they are softened, 5 to 6 minutes. Carefully add the lentils and the boiling water. Add the tomato paste and 1/4 teaspoon of crushed red pepper to the soup and season with salt and black pepper. Cook the soup over moderate heat until the lentils are tender, about 20 minutes.
Meanwhile, in a small skillet, heat the remaining 1 tablespoon of olive oil until shimmering. Add the minced garlic and cook just until fragrant, about 30 seconds. Add the ham and a generous pinch of crushed red pepper to the skillet and cook for 30 seconds. Add the arugula and cook just until wilted, 30 seconds longer. Ladle the lentil soup into large bowls, garnish with the ham and arugula and serve.