Preheat the oven to 400°F. Scrub your potatoes clean and pat them dry with a kitchen towel. Use the tines of a fork to poke each potato in several places all around.
Place the potatoes on a baking sheet and cook in the oven until easily pierced with a fork, about 1 hour. Remove from the heat and set aside to cool.
Place the chopped carrots in a saucepan and add water until just covered. Bring to a boil on high heat and reduce to simmer over medium heat until tender, about 20 minutes. Drain and set aside.
Once the baked potatoes are cool enough to handle, peel them. The skins should slip off easily enough by hand, or you can use a small paring knife to help things along.
Cut the peeled, cooked potatoes into bite sized chunks, then set aside.
Add the vegan butter to a stock pot and cook over medium heat until just melted.
Add the diced onion and saute, stirring often, until tender and translucent, about 10 minutes.
Sprinkle the flour over the cooked onions and stir until well combined. Continue cooking--stirring constantly--until the raw flour smell dissipates, about 2 minutes.
Turn the heat up to high, then slowly add the non-dairy milk and vegetable broth. Stirring or whisking constantly to prevent clumps.
Bring the mixture to a boil, then reduce the heat to medium to activate the flour's thickening power. . If the soup seems too thick, add more vegetable stock, a little at a time, to adjust the consistency.
Stir in the paprika, thyme, salt, and black pepper.
Gently stir in your chopped cooked potatoes and carrots.
Once the veggies are warmed through, this soup is ready to serve! Garnish with black pepper, if desired.