In a mini food processor or blender, combine first 7 ingredients. Blend until garlic and ginger are chopped and well combined. If you don’t own a mini food processor or blender, simply mince the garlic and ginger as fine as possible and give everything a good whisking. Set aside.
Heat a deep sauté pan (something with sides) over medium-high heat and add the coconut and sesame oils. Add broccoli florets and sauté for about 6 - 10 minutes or until broccoli starts to char a bit on the outside. Lower heat to medium and add cabbage, green onions and lime juice and cook until cabbage just begins to soften, about 10 minutes.
In a separate pot (I used a medium sized pot, not a large one like you normally need for a box of pasta), boil water for the udon noodles. Please note that I did not salt the water for these noodles. The soy sauce in the vinaigrette will season everything nicely so you don't need additional salt here. Cook udon noodles according to package directions (I cooked mine for five minutes). Drain.
Off the heat, add udon noodles back into the pot you cooked them in. Add half of the vinaigrette mixture, stirring to combine well. Let them rest for five minutes.
After resting, add veggie mixture, remaining vinaigrette and chopped cilantro to the noodles. Stir to combine.
Serve in bowls topped with peanuts or almonds and fresh cilantro sprigs.