To make the soup: In a medium sauté pan (it might seem too big right now but later we'll use it to cook the broccoli), set over medium heat, melt the butter. Add the shallot and garlic and cook for 1 to 2 minutes, until softened. Add it to the blender and set aside.
Prepare an ice bath for the asparagus and set aside. In a small saucepan filled with a few inches of water, set over medium heat, add a few pinches of salt and the baking soda. When the water reaches a simmer, drop the asparagus in the water and cook for 2 minutes, until tender. Drain and transfer to the ice bath. Drain again and press out any water in the asparagus. Transfer to the blender along with the chicken stock. Blend until very smooth, 1 to 2 minutes.
If you like, you can run it through a sieve. I liked it nice and thick. Add the creme fraiche and a teaspoon of salt. Give it one more mix and adjust the salt to taste. Transfer back to the saucepan and warm over low heat. Ladle into bowls and top with a dollop of creme fraiche and chives.
To make the grilled cheese: In the sauté pan you cooked the onions and garlic in, add a teaspoon of olive oil (if needed), and set over medium heat. Add the broccoli and sprinkle with salt and crushed red pepper. Cook for 1 to 2 minutes, until tender and bright green.
Assemble the grilled cheese by dividing the cheese amongst slices of bread and topping with the broccolini. Add the second slice of bread. Wipe out the skillet and place over medium heat. Melt a tablespoon of butter in the pan and cook the grilled cheeses on each side for 1 to 2 minutes, covering the pan as needed. Repeat with the second grilled cheese.
Slice the grilled cheeses at a diagonal. Serve grilled cheese with the asparagus soup.