Cook the Mushrooms and Onions:
Heat oil or butter in a large saucepan over medium-high heat.
Add mushrooms and onions, and cook until the mushrooms release their liquid and it evaporates, about 10-15 minutes.
Add Garlic, Thyme, and Flour:
Mix in the garlic, thyme, and flour. Cook, stirring, for 1 minute.
Deglaze and Simmer:
Add the white wine to deglaze the pan, scraping up any browned bits from the bottom.
Add the chicken broth, tortellini, and chicken. Bring to a boil, then reduce heat and simmer for 10 minutes, until the tortellini is tender and the chicken is cooked.
Shred the Chicken:
Remove the chicken, dice or shred it, and mix it back into the soup.
Finish the Soup:
Stir in the heavy cream, Dijon mustard, and Asiago cheese. Let the cheese melt.
Season with salt and pepper to taste, then remove from heat.