Preheat the oven to 375°F.
Melt the tablespoon of butter in a small skillet over medium heat. Add the 3/4 cup diced onion and sauté, stirring occasionally, until the onion is tender and lightly browned, about 5 minutes. Transfer the cooked onion to a small bowl and set aside to cool slightly.
Trim the asparagus by bending each spear to snap off the woody ends. Place the asparagus spears in a shallow microwave-safe bowl, add a little cold water, and microwave 1–3 minutes until crisp-tender. Drain and transfer the spears to ice water to stop the cooking. Drain and pat dry, then set aside.
In a large bowl, whisk the 3 large eggs with the 3 tablespoons heavy cream and whole milk mixture until combined. Stir in the cooked onions, the 2 diced chicken sausages, and half of the 1 1/4 cups grated Dubliner cheddar cheese. Season with salt and pepper to taste and mix until evenly combined.
Place the unbaked 10-inch tart shell on a baking sheet. Pour the egg and sausage mixture into the tart shell, spreading evenly.
Arrange the asparagus spears on top of the filling and sprinkle with the remaining grated cheese.
Cover the tart loosely with foil and bake at 375°F for 15 minutes. Remove the foil and continue baking 5–10 minutes more, until the center is set (a gentle jiggle only) and the top is lightly golden.
Remove the quiche from the oven and let it cool for 10 minutes before slicing and serving.