Combine shrimp, mint, garlic, lemon juice, and 1/4 cup olive oil in a medium bowl. Cover, and refrigerate for 1 hour.
Meanwhile, whisk together lime juice, fish sauce, chili sauce, vegetable oil, and sugar. Let stand until ready to use.
Place noodles in a large bowl. Bring a pot of water to a boil; pour over noodles. Let stand until softened, 6 to 8 minutes. Drain, and rinse under cold water.
Remove shrimp from marinade, and season with salt and cayenne. Heat remaining 2 tablespoons olive oil in a large skillet over medium-high heat. Add shrimp in a single layer. Cook until golden and cooked through, about 1 minute per side.
Divide noodles among 4 serving bowls. Top with shrimp, cucumber, and chile. Drizzle lime juice mixture over the top, and serve with lime wedges.
Glass noodles, also known as cellophane noodles, are thin strands of dried mung-bean starch. They are available online and at Asian markets, as are chili sauce and Thai chiles.