Place the cut potatoes in a large pot and cover with cold water by about 1 inch. Season the water generously with salt. Bring to a boil over high heat and cook for 15-20 minutes, or until the potatoes are tender when pierced with a fork but not falling apart. Drain them well in a colander.
While the potatoes are boiling, heat the olive oil in a large, deep skillet or Dutch oven over medium heat. Add the chopped onion and cook, stirring occasionally, until softened, about 5 minutes.
Add the minced garlic and cook for 1 minute more until fragrant.
Stir in the all-purpose flour, dried oregano, dried thyme, smoked paprika, and red pepper flakes. Cook, stirring constantly, for 1 minute.
Gradually whisk in the milk until smooth, then whisk in the vegetable broth. Stir in the chopped sun-dried tomatoes.
Bring the sauce to a simmer and let it cook for 3-5 minutes, stirring occasionally, until it has thickened. Season to taste with salt and pepper.
Gently add the drained potatoes to the skillet with the sauce. Stir carefully to coat the potatoes. Allow it to simmer for 5 minutes so the potatoes can absorb some of the sauce.
Remove the skillet from the heat. Add the grated cheese, fresh basil, and lemon juice. Stir gently until the cheese is melted and everything is well combined. Serve immediately.