Heat the saucepan over medium high.
Add 1 Tbsp. of oil and the butter to the pan.
Once the butter is melted and sizzling, add the shallot and garlic,.
Cook until shallots are starting to caramelize while stirring, about 4 minutes.
Add all but 2 ounces of mushrooms and the salt to the pan, and stir to combine.
Sauté until mushrooms have released excess moisture and brown, about 7 minutes.
Add rosemary and stir to combine.
Add white wine, stir and scrape the pan surface to deglaze the browned bits.
Add water and chicken bouillon, stir to dissolve.
Add half and half and black pepper.
Bring up to a boil and remove from heat.
Using an immersion blender carefully blend until smooth and pureed.
Return to saucepan and bring to a boil on high heat.
While stirring rapidly, drizzle in corn starch slurry to thicken.
Turn heat to low.
While the soup simmers, heat small skillet over high heat.
Add remaining oil to skillet, once heated through, add remaining mushrooms.
Sauté mushrooms until tender, about 3 minutes.
Remove small skillet from heat.
Add asiago cheese to the soup, stir until mixed in.
Remove soup from heat and portion into bowls.
Garnish each bowl with sautéed mushrooms.
