Start by prepping the potatoes: peel four of them (if you're using larger potatoes, only peel two), then set the peels aside and cube both the peeled and the unpeeled potatoes.
Add the butter to a medium to large nonstick pot and heat over medium-high heat. While the butter is melting, add the potatoes to the pot and sauté for 5 minutes.
During those 5 minutes, prep the green onions and thyme: wash the green onions, pat them dry, and cut off the darkest green parts, then set those aside (you'll need them to garnish the soup.) Cut off and discard the ends (with the roots), then slice or chop the remaining white and light green parts of the green onions. Remove the thyme leaves from the stems and discard the stems.
Add the (white and light green parts of the) green onions, thyme leaves, garlic powder, onion powder, celery salt (if using) and 1/2 of the dijon mustard to the pot. Mix and sauté for 3 more minutes. Add the white wine to deglaze for 2 additional minutes.
Add the vegetable broth and bring to a boil over high heat. Once it's boiling, lower the heat back to medium-high and cook for 10 minutes.
Check if the potatoes are fully cooked. Next, transfer the soup from the pot to the pitcher of your blender, add the other 1/2 of the dijon mustard to the blender and blend on high until smooth (about 30-60 seconds.) Return the soup to the pot and add sea salt and pepper to taste.