Bring a 4-qt. saucepan of water to a boil; add salt. Add shrimp; poach until just cooked, about 2 minutes. Drain shrimp and transfer to a plate; let cool. Finely chop shrimp; set aside in a large bowl. Melt 1 tbsp. butter in a 2-qt. saucepan over medium heat. Add thyme, carrots, leeks, fennel, and celery and cook, stirring often, until soft, about 10 minutes. Add 1 tsp. ginger and cook for 2 minutes. Add 3 tbsp. water, cover partially, and cook until liquid has evaporated, 4-5 minutes; set aside to let cool. Remove thyme and transfer vegetables to bowl containing chopped shrimp; season shrimp mixture with salt and pepper and stir to combine. Set aside.
Using a pastry brush, brush a wonton wrapper with egg and put 1 tbsp. shrimp mixture into center of wrapper. Top with another wonton wrapper and press edges to seal; trim ravioli with a 2 1⁄2" round cookie cutter to produce a round dumpling (or leave square, if you like). Repeat with remaining wonton wrappers and shrimp mixture to make 16 ravioli in all. Transfer ravioli to a parchment paper-lined baking sheet, cover with plastic wrap, and refrigerate.
Bring 4 cups salted water to a boil in a 6-qt. saucepan over high heat. Add spinach and cook for 1 minute; drain, squeezing out as much liquid as possible, and set aside. Heat remaining ginger, vermouth, and 1 1⁄2 cups water in a 1-qt. saucepan over medium-high heat; cook until reduced to 2 cups, about 10 minutes. Remove from heat and whisk in remaining butter, 1 tbsp. at a time, until incorporated. Set a fine-mesh sieve over a 1-qt. saucepan and strain sauce, discarding solids. Add tomatoes and season with salt and pepper; keep warm.
Divide spinach between 4 warmed serving plates. Bring a large pot of salted water to a simmer over medium heat. Add ravioli and cook until filling is hot, 4-5 minutes more. Drain ravioli and place 4 on each bed of spinach. Spoon ginger broth over ravioli and garnish with chives.