Place the lentils, stock, water, tomato paste, & a little salt & pepper into a saucepot over high heat. Bring to a simmer, stirring occasionally, cook for about 20-30 minutes until tender.
If using Bacon and sausage: Heat the bacon in a large sauté pan over med-low to medium heat. After a couple of minutes, add the sausage. Cook, stirring occasionally, for about 8-10 minutes until most of the fat has rendered off. Remove the bacon pieces from the pan with a slotted spoon, reserving the fat in the pan.
If you are not using bacon or sausage. Heat 1 Tbsp of oil or butter in a medium saute pan. If you are using bacon, just use the same pan with the bacon fat in it.
Raise the heat to medium heat, add in the onion, carrot, celery, thyme, & some salt & pepper. Sauté for about 6-8 minutes until the veggies are starting to brown. Reduce the heat to low & stir in the lentils, vinegar & garlic. Remove from the heat, stir in the parsley and arugula, & season with salt & pepper.
