Prepare an outdoor charcoal grill for medium-high heat.

Place dry spaghetti on the hot charcoal grill, and cook, tossing often with tongs until the pasta is toasted to a light brown, about 4 minutes. Remove and set aside.

Add olive oil to a pan, and set over medium heat. Add garlic and sauté until fragrant, 30 seconds to 1 minute. Add tomato paste and sauté for about 1 minute; pour in broth.

Raise heat to medium-high and bring broth to a simmer. Add salt and grilled spaghetti, and use tongs to move spaghetti back and forth until it softens enough to be covered by the broth.

Cook, stirring, until spaghetti is as tender as you like; it will take a few minutes longer than package directions indicate. If pan gets too dry, add another splash of broth.