Turn on a multi-functional pressure cooker (such as Instant Pot) and select Saute function. Add 1 tablespoon butter and allow to melt. Add freekeh, carrots, celery, onion, mushrooms, and garlic; saute for 3 minutes. Pour in broth.
Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 10 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure using the natural-release method according to manufacturer's instructions, about 20 minutes. Unlock and remove the lid. Season soup with salt, poultry seasoning, and thyme. Stir to combine.
Melt remaining 6 tablespoons butter in a skillet over medium-high heat and stir in flour to create a paste. Whisk in milk until sauce is smooth and thickened, 3 to 5 minutes. Pour into the soup and stir to combine. Serve immediately.