Chop the onion finely. In a small stockpot, melt 2 tbsp butter over medium heat. Add the onion and a couple of pinches of salt and pepper. Cook stirring often, until softened, about 5 minutes.
Stir the peas, edamame, 1 1/2 cups milk and the thyme into the pan. Cover and simmer on medium-low heat for 5 minutes.
Remove the thyme sprigs from the soup using a pair of tongs; discard.
Puree the soup and the remaining 1/2 cup milk with an immersion blender (or let cool and puree in a regular blender) until smooth, about 5 minutes. Season with salt and pepper. Cover and keep warm over very low heat, stirring occasionally.
Whip the heavy cream with the egg beater until stiff.
Cut the crusts off the bread. Assemble 4 sammies with the cheese, cutting it to fit the bread.
Melt 1/2 tbsp butter over medium-low heat in a large skillet. Add 2 sammies; cook until golden on the bottom (use a spatula to take a peek to see if they're done), about 5 minutes.
Add 1/2 tbsp butter to the pan and flip the sammies. Cook until the bread is golden and the cheese is melted, 3 minutes. Cut each sandwich into 3 sticks. Repeat steps 7 and 8 with the remaining butter and sammies.
Ladle the soup into bowls, dollop with the whipped cream and top with the chives. Serve with the grilled cheese sticks.