Hydrogelled emulsions (HEs) of linseed oil and pea protein (PP) were produced with four levels (0, 5, 7.5, and 10%) of raspberry extract obtained by a green extraction technique (microwave hydrodiffusion and gravity—MHG). HEs were applied in burgers to replace 50% of pork backfat content. The products’ technological, nutritional, oxidative, microbiological, and sensory properties were evaluated. Besides reducing the fat level by approximately 43%, the reformulation reduced the n-6/n-3 PUFA ratio to healthy levels, decreased the diameter reduction by 30%, and increased the cooking yield by 11%. Including 7.5 and 10% of raspberry extract in the HEs decreased the oxidative defects caused by the enrichment of the burgers with omega-3 fatty acids. In addition, the raspberry extract did not cause alterations in the mesophilic aerobic count and the burgers’ sensory profile.