1. Preheat the oven to 350 degrees. Line a baking sheet with foil.
2. Add potatoes to a large pot and fill with cold water. Season liberally with salt and bring to a boil over medium-high heat. Cook for 15-20 minutes or until the potatoes are fork tender.
3. Meanwhile, add the butter and onions to a small saucepan over medium-low heat. Cook for 20 minutes, stirring occasionally, until the onions have caramelized.
4. Add the olive oil to a skillet over medium-high heat and once hot, cook the sausages for 5-7 minutes or until brown on all sides. Transfer to the prepared baking sheet and finish cooking in the oven while you finish the rest of the meal.
5. Add the garlic and thyme to the caramelized onions and cook for 1-2 minutes. Add the beer to the saucepan and stir with a wooden spoon, dislodging any of the stuck bits of onion.
6. Stir in the mustard and once combined, sift the cornstarch over the mixture and whisk in the beef broth, stirring occasionally until the mixture reduces and thickens.
7. Meanwhile, drain the potatoes and add the milk, horseradish, salt and pepper. Mash the potatoes until smooth.
8. Plate the potatoes, followed by 2 of the sausages and a heaping spoonful of the onion-mustard gravy on top.