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Evolved stars are crucial pieces to understand the dependency of the planetformation mechanism on the stellar mass and to explore deeper the mechanisminvolved in star-planet interactions. Over the past ten years, we havemonitored about 1000 evolved stars for radial velocity variations in search forlow-mass companions under the Penn State - Torun Centre for Astronomy PlanetSearch program with the Hobby-Eberly Telescope. Selected prospective candidatesthat required higher RV precision measurements have been followed with HARPS-Nat the 3.6 m Telescopio Nazionale Galileo under the TAPAS project.  We aim to detect planetary systems around evolved stars to be able to buildsound statistics on the frequency and intrinsic nature of these systems, and todeliver in-depth studies of selected planetary systems with evidence ofstar-planet interaction processes. For HD 5583 we obtained 14 epochs of preciseRV measurements collected over 2313 days with the Hobby-Eberly Telescope (HET),and 22 epochs of ultra-precise HARPS-N data collected over 976 days. For BD+152375 we collected 24 epochs of HET data over 3286 days and 25 epochs of HARPS-Sdata over 902 days.  We report the discovery of two planetary mass objects orbiting two evolvedRed Giant stars: HD~5583 has a m sin i = 5.78 M$_{J}$ companion at 0.529~AU ina nearly circular orbit (e=0.076), the closest companion to a giant stardetected with the RV technique, and BD+15~2735 that with a m sin i= 1.06M$_{J}$ holds the record of the lightest planet found so far orbiting anevolved star (in a circular e=0.001, 0.576~AU orbit). These are the third andfourth planets found within the TAPAS project, a HARPS-N monitoring of evolvedplanetary systems identified with the Hobby-Eberly Telescope.

Tracking Advanced Planetary Systems (TAPAS) with HARPS-N. III. HD 5583 and BD+15 2375 - two cool giants with warm companions