Combine 4 cloves roughly chopped garlic with salt, pepper and olive oil. Add rosemary leaves and thyme leaves. Smear around the bowl and squeeze in your fist to bruise and combine. Smear this paste all over the short ribs on a baking sheet. Place sheet in oven and roast 2 1/2 hours, until the meat is falling off the bone.
Heat 2 T olive oil over medium-high heat in a large skillet. Add mushrooms and sear 3-5 minutes until browned. Add shallots and garlic; cook until garlic is fragrant, 1-2 minutes. Season with salt and pepper. Move pan away from open flame and add cognac to deglaze; return pan to heat and scrape any browned bits off the bottom of the pan. Add heavy cream. Bring to a boil, reduce to a simmer and cook until reduced by half. Turn off heat and add mustard and sour cream; stir well. Adjust seasoning with salt and pepper.
While sauce cooks, prepare noodles according to package directions. Drain and toss with butter and a little pepper while hot.
Remove meat from bones and slice crosswise, 1/2" thick. Serve noodles, topped with sauce, then place meat on top. Garnish with parsley and green onions.