Make the cupcakes: Heat oven to 350°F (176°C). Grease a muffin tin with nonstick spray or line it with cupcake liners and set aside. In a large bowl, beat the egg for a few seconds. Gently stir in the grated carrots and oil. Add the rest of the ingredients and stir to combine, about 10 stirs (be careful not to over-mix). Fill cupcake wells about 2/3 full. Bake for no more than 18-20 minutes or until golden brown. Remove from oven and allow to cool for at least 10 minutes before frosting.
Make the frosting: beat the butter, cream cheese, and vanilla until fluffy. Gradually add the powdered sugar and cream until light and fluffy. Once the cupcakes have cooled, frost them with cream cheese frosting.
Prep the ingredients per the instructions above.
Assemble the salad ingredients into a bowl. Wait to dress the salad until ready to serve.
Heat the oil in a skillet over medium heat until shimmering. Arrange sausages in pan and cook, turning once, until browned on 2 sides, about 5 minutes. Move sausages to 1 side of the skillet. Add onions, evenly distributing around the bottom of the pan, and nestle sausages on top. Add the water and immediately cover. Cook, turning sausages once until they register between 160°F (71°C) and the onions have softened about 10 minutes.
While sausages cook, whisk broth, soy sauce, mustard, thyme, and rosemary in a bowl.
Make the potatoes: Add the potatoes to a pot and cover them with water over high heat. Bring to a boil and cook until tender, about 10 to 12 minutes. Melt the butter in a saucepan over medium heat, then stir in the cream and remove from the heat.
Transfer the sausages to a plate and tent with aluminum foil. Make sure onions are spread evenly; cook without stirring until beginning to brown, about 5 minutes (if onions have not been browned, increase heat to medium-high). Continue to cook, stirring occasionally, until onions are tender and well browned, 3 to 5 minutes longer. Stir in sugar, salt, and pepper and cook for 1 minute.
Add broth mixture, increase heat to medium-high, and bring to a boil. Cook, scraping up any browned bits from the bottom and sides of the skillet and stirring back into sauce until sauce is slightly reduced about 5 minutes. Combine cornstarch with 1 tablespoon of water in a small bowl. Whisk cornstarch mixture into sauce and cook until sauce is glossy and has a creamy consistency, about 2 minutes.
Remove from the heat, and whisk in butter. Stir in vinegar and season with salt and pepper to taste.
Once the potatoes are tender, drain well, then mash with a fork or masher. Mix in the butter cream mixture until fully combined. Season with salt and pepper to taste.
Warm the roll in the oven for 2 to 3 minutes. Dress the salad.
Serve the sausage over the mashed potatoes, and spoon the gravy over top. Garnish with parsley. Serve with salad, roll, and sparkling water. Enjoy with cupcake.
Save the remaining cupcake in a container with a tight-fitting lid.