Preheat the oven broiler to high and position a rack about 6–8 inches from the heat.
Pat the salmon dry and season both sides with the teaspoon of salt and teaspoon of pepper; place the fillet on a baking sheet.
Melt 2 tablespoons of the butter in a small saucepan over medium heat, add the minced garlic, and cook briefly until fragrant, about 30–60 seconds; remove from heat.
Brush some of the garlic butter over the top of the salmon.
Broil the salmon until the top is opaque and the fish flakes easily with a fork, about 8–10 minutes for a 1-inch-thick fillet (adjust time for thickness).
Remove the salmon from the oven and brush with the remaining garlic butter before serving.
While the salmon cooks or after removing it, heat 1 tablespoon butter and 1 tablespoon olive oil in a small saucepan over medium-low heat.
Add the sliced shallots and a pinch of salt and cook, stirring occasionally, until they turn translucent, about 5 minutes.
Stir in 1 tablespoon brown sugar and continue cooking until the shallots caramelize, about 10 more minutes.
Stir in the corn kernels and the diced cherry peppers (or pimentos) and cook for 1–2 minutes more until warmed through.
Serve the broiled garlic butter salmon topped with the caramelized shallot corn relish.