Combine sugar, water, and salt in a large bowl, stirring until sugar and salt dissolve. Add pork; cover and brine 2 hours to overnight in refrigerator.
Preheat oven to 350°.
Heat oil in a large nonstick skillet over medium heat. Add sliced fennel, onion, and 2 garlic cloves; cook until golden brown (about 20 minutes), stirring frequently. Add 1/4 cup broth; cook until liquid evaporates, scraping pan to loosen browned bits. Stir in 1/4 teaspoon pepper. Add remaining broth; bring to a boil. Remove from heat.
Combine bread and fennel mixture. Stir in cheese. Spoon mixture into an 11 x 7-inch baking dish coated with cooking spray. Bake at 350° for 1 hour and 10 minutes or until golden brown.
While bread pudding bakes, rinse pork and pat dry. Combine fennel seeds, 2 garlic cloves, and 1/4 teaspoon pepper; rub evenly over pork.
Heat a large ovenproof skillet over medium-high heat. Coat pan with cooking spray. Add pork to pan; cook 5 minutes, browning on all sides. Place skillet in oven; bake at 350° for 40 minutes or until a thermometer registers 140° (slightly pink). Let stand 10 minutes before serving. Serve bread pudding with pork.