Bring 5 cups salted water to a boil in a 4-qt. saucepan over high heat. Gradually whisk in grits; reduce heat to medium-low, and cook, whisking constantly, until thickened and tender, about 5 minutes. Remove from heat and stir in cheddar, half-and-half, butter, cayenne, garlic paste, and season with salt and pepper. Pour grits onto a serving platter.
Heat a cast iron grill pan over medium-high heat. Grill ramps in 2 batches until leaves are wilted and ramps are tender and slightly charred, about 5 minutes. Place over grits, drizzle with oil and season with salt and pepper.