Heat oven to 400°F. Toast the cut bagel faces under the broiler for 1 to 2 minutes, then return the oven to bake.
Flake drained tuna into a bowl and fold in artichokes, roasted pepper, mayonnaise, lemon juice and black pepper. Keep the mixture chunky but evenly bound.
Divide tuna mixture among the four bagel bottoms on a sheet pan. Cover with half the Fontina and bake for 8 to 10 minutes until the filling reaches 165°F.
Put remaining Fontina on the top halves and add them cut side up to the pan. Broil all halves for 1 to 2 minutes until the cheese on both sets melts.
Turn the top halves over the filled bottoms, rest for 1 minute and cut the closed melts in half.